The Classical Mass Action Paradigm and an Application to Pest Control

Utah State University DigitalCommons@USU

All Graduate Plan B and other Reports Graduate Studies

5-2016

Modeling of Mating Encounters: The Classical Mass Action Paradigm and an Application to Pest Control

Katherine R. Snyder Utah State University

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Recommended Citation Snyder, Katherine R., "Modeling of Mating Encounters: The Classical Mass Action Paradigm and an Application to Pest Control" (2016). All Graduate Plan B and other Reports. 807. https://digitalcommons.usu.edu/gradreports/807

This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Plan B and other Reports by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. MODELING OF MATING ENCOUNTERS: THE CLASSICAL MASS ACTION PARADIGM AND AN APPLICATION TO PEST CONTROL

Katherine R. Snyder

A thesis submitted in partial fulﬁllment of the requirements for the degree

MASTER OF SCIENCE

Industrial Mathematics

Approved:

Luis F. Gordillo Brynja Kohler Major Professor Committee Member

Edward W. Evans Committee Member

UTAH STATE UNIVERSITY Logan, Utah

2016 ii

Abstract

Modeling of Mating Encounters: The Classical

Mass Action Paradigm and an Application

to Pest Control

Katherine R. Snyder, Master of Science

Utah State University, 2016

Major Professor: Dr. Luis F. Gordillo Department: Mathematics and Statistics

The growth of a two-sex population is undoubtedly dependent upon the dynamics of its mating encounter rates. Encounter rates are influenced by several factors that affect the modeling of a population. In this work we first look at the law of mass action as applied to mating encounter rates. We review the underlying assumptions of mass action and present a derivation using dimensional reduction and simulated data. This approach led to a revised proportionality constant that seems to produce results in better accord to experimental data than does the original constant. We also explore numerically how random fluctuations on the revised constant affect the conditioned time to extinction of a two-sex population subject to reproductive Allee effect. Next we present an application to pest management. We consider the effectiveness of chemosterilant induced infertility in a rat population and present qualitative predictions for population behavior when subject to this fertility control. We look at the dynamics of both a spatially isolated rat population and a population with multiple rat communities where individual movement between patches occur.

(57 pages) iv

Public Abstract

Modeling of Mating Encounters: The Classical

Mass Action Paradigm and an Application

to Pest Control

Katherine R. Snyder, Master of Science

Utah State University, 2016

Major Professor: Dr. Luis F. Gordillo Department: Mathematics and Statistics

This approach led to a new proportionalty constant that seems to produce results in better agreement to experimental data than does the original constant. We also explore numerically how random changes on the revised constant aﬀect the conditioned time to extinction of a two-sex population subject to limited reproduction at low densities.

Next we present an application to pest management. We consider the eﬀectiveness of a chemical bait that induces failed reproduction within a rat population and present qualitative predictions for population behavior when subject to this fertility control.

We look at the dynamics of both a spatially isolated rat population and a population with multiple rat communities where individual movement between patches occur.

(57 pages) v

Acknowledgements

I would like to thank my teachers and colleagues at USU for helping me to get where I am today. I would especially like to thank Drs. Luis Gordillo and Brynja

Kohler for sparking my interest in math biology and encouraging me to continue to this point in my education. I would also like to thank my family for their unfailing love and support through everything. Thank you Mom and Dad for all that you’ve done to help me along the way and thank you to my wonderful husband Chris who has been with me and supported me each day since I started this journey.

Katherine R. Snyder vi

Contents

Abstract...... iii

Public Abstract...... iv

Acknowledgements...... v

List of Figures...... vii

1 Introduction...... 1

2 Using the Law of Mass Action to Model Mating Encounters..... 4 2.1 Introduction...... 4 2.2 Dimensional reduction for the law of mass action...... 5 2.2.1 Pair formation...... 8 2.3 Eﬀects of environmental stochasticity...... 10 2.4 Discussion...... 13

3 A Model for Rat Management...... 15 3.1 Introduction...... 15 3.2 Population dynamics in one patch...... 16 3.2.1 Mating encounters...... 16 3.2.2 Female and male sterilization...... 17 3.2.3 Numerical experiments...... 19 3.3 Coupling patches...... 20 3.3.1 Movement between patches...... 20 3.3.2 Territoriality...... 23 3.3.3 Numerical experiments...... 24 3.4 Using GIS to determine patches...... 25 3.5 Conclusions...... 28

Appendices...... 36

A Pair Formation Data...... 37

B NetLogo Code for Pair Formations...... 49

C Matlab Code for Pest Control Model...... 52

D Python Code with GIS to Predict Rat Populations...... 55 vii

List of Figures

2.1 Figure 2.1...... 9 2.2 Figure 2.2...... 12

3.1 Figure 3.1...... 18 3.2 Figure 3.2...... 19 3.3 Figure 3.3...... 21 3.4 Figure 3.4...... 22 3.5 Figure 3.5...... 26 3.6 Figure 3.6...... 27 3.7 Figure 3.7...... 29 1

Chapter 1

Introduction

The dynamics of a population are largely dependent upon its ability to reproduce. For reproduction to occur in a two-sex population an individual must encounter an individual of the opposite sex and mate. The mechanisms behind these encounters may vary greatly between species. In insects it is common for a chemical attraction to aid in the encounter between mates. Moths (Lepidoptera), Coleoptera, and Diptera are among those whose females emit a pheromone to attract males. Some cases of male emitted pheromones have been discovered within Lepidoptera and other species use a dual system where both males and females emit pheromones to attract a mate [5]. Other types of sensory signals such as bright colors, complex sounds, horns, eye stalks and feathers can also be used to attract mates [4]. Also important in population dynamics of a species are systematic failures of mating encounters.

Systematic failures in mating encounters can occur as a result of individuals failing to find a mate. Failure to find a mate is seen in many species at low densities and is known as an Allee effect. There have also been methods of pest management which confuse individuals and reduce the number of mating encounters. For instance, pseudo- pheromones for the apple codling moth released by humans can cause the males to search for a female in an area where there are no females and thus reduce the number of male-female mating pairs that are formed [36]. This helps pest managers reduce a pest population but not all failed mating encounters are as useful. When the sterile insect technique is used and sterilized males are introduced to a population, failure for females to mate with the irradiated (sterile) males decreases the effects of the technique in managing the pest population [29]. The ways in which mating encounters occur and fail to occur are important when modeling the population dynamics of a system. 2

Why is modeling population dynamics important? Understanding the dynamics of a population is important for many people. Farmers and landscapers care about managing pests that harm plants and crops, cities care about managing pests that cause damage and spread diseases, hunters and fishermen care about the surplus of a population they hunt, and wildlife conservationists care about saving endangered species. In these and other situations modeling a population can provide answers to how a system will grow (decline) and help identify the most effective way of managing the population. A better understanding of how mating encounters are modeled can help increase the effectiveness of a model in answering the questions asked for managing a population.

Our research here has two parts. The first is an endeavor to better understand one of the most common forms of modeling mating encounters, the law of mass action. We study the law of mass action and the appropriateness of its use in estimating encounter rates. The law of mass action was first used to model interaction rates by A.G. McKendrick for describing interactions among susceptible and infected individuals in epidemiological contexts [18][27]. After its use in Alfred Lotka’s Elements of Physical Biology [26] to justify the encounter term in his predator-prey system of differential equations it has become common in mathematical ecology for modeling interactions between individuals of different groups, including mating encounters between individuals of opposite sex. To better understand the role of mass action when used to estimate mating encounters we derive the law using dimensional reduction along with simulated data. We consider the underlying assumptions of mass action and suggest a modifica- tion for the use of mating encounters with more realistic assumptions. Our approach seems to offer results better in agreement with experimental data. We also are able to explore how numerically random fluctuations of the suggested proportionality constant may affect the time to extinction of a two-sex population subject to reproductive Allee effect.

In the second part of our research we look at an application to rat management. We model the population dynamics of rats when subject to fertility control. We begin by looking at a closed population of spatially isolated rats and approximate population behavior when fertility control in the form of a sterilizing bait (chemosterilant) is introduced. We then expand to consider multiple rat communities with individual movement between patches. A community of rats will behave significantly different than an isolated population due to territorial behavior [34]. Movement between patches is modeled to 3 capture this territorial behavior and show how the effectiveness of the chemosterilant is affected by changes in density and in how interactions between rat communities occur. 4

Chapter 2

Using the Law of Mass Action to Model Mating Encounters

This paper has been submitted to a peer reviewed journal under the title:

The law of mass action in two-sex population models: encounters, mating encounters

and the associated numerical correction

K. Snyder, B. Kohler, Luis F. Gordillo

2.1 Introduction In chemistry, the law of mass action states that the rate of a reaction is proportional to the product of the concentrations of the reactants. Since Alfred Lotka pioneered its use in [26] to justify the encounter term in his predator-prey system of differential equations, the law has become ubiquitous in mathematical ecology for modeling the interactions between individuals of different groups. Lotka’s arguments for the use of mass action in biological encounters were motivated by the analogy to the kinetic theory of gases (ideal gas models) and have been applied to describe a variety of phenomena, including fertilization kinetics, search theory and mate finding, see [19, 37] for thorough reviews. Remarkably, before Lotka, the chemical law of mass action was used for the first time by A.G. McKendrick for describing the interactions among susceptible and infected individuals in epidemiological contexts [18, 27].

In the theory of molecular collisions in gases, the collision frequency among particles of different types is expressed in terms of the velocities and radii of particles. The quantitative law is deduced directly from geometrical abstractions using the mean of a Poisson process that models the number of collisions a particle receives from others and where simultaneous collisions are allowed [19, 20]. At relatively low population densities, the same idea is used as a phenomenological approach to approximate the encounter rates between males and females, with the birth rate taken proportional to the product of their densities, [2]. We remark that two-sex population models become relevant when 5 sexual dimorphism in vital rates is present, which has been observed in several species [7]. The modeling of encounters that lead to reproduction, however, requires simultaneous encounters to be ruled out, otherwise the rate at which new offspring appear would be overestimated (in addition to the fact that simultaneous mating encounters are not biologically sound). Thus, we differentiate between counting encounters, possibly simultaneous, and counting mating encounters, which are understood here as the formation of female-male pairs from which offspring are successfully produced. Here we present a correction for the constant used in the mass action term that accounts for the pair formation. The motivation for this work is our interest in the theoretical modeling of ephemeral (short-lived) mating encounters for insects.

First we build a functional relation among the variables by using dimensional reduction and simulated data of individuals’ movement. This allows us to approximate the value of the proportionality constant for the mass action law with relative precision in comparison to theoretical results. Then we generate new data through computer simulations that only count female-male pairs and with this we approximate the new value for the constant. Finally, we used the new constant in the mass action law to explore the effects of environmental stochasticity on the conditioned time to extinction for a population model, via the variability on encounter rates. The stochastic model used has a deterministic skeleton that approximates mass action at low densities and thus shows a reproductive Allee effect [10]. Our simulations confined the effects of encounter rate variations to demographic parameters. This elementary example emphasizes the importance of having reasonable approximations for the nonlinear term that models encounters: it demonstrates how variability in the environment could play an essential role in regulating the time to extinction distribution through induced random fluctuations on the vital rates.

2.2 Dimensional reduction for the law of mass action We start by deriving the classical form of the law of mass action, which accounts for encounters (not pair formation) and assumes that individuals’ movement is done in two spatial dimensions, i.e. the total variation in the displacement of each individual on the plane is much larger than changes made in their altitude. Thus, we consider a two- sex population moving over a two dimensional terrain with area A under the following hypotheses: 6

• the velocity, v, is constant and the same for both sexes,

• individuals move independently from each other,

• movement is in straight lines,

• the initial direction of movement for each individual is independently chosen at random from the interval [0, 2π) and remains unchanged at each time step,

• individuals’ sizes are negligible.

Let nm and nf be the number of males and females, respectively, and c the average number of encounters that one female has with males during the observation time t. We hypothesize that c is related to (i) the velocity v, (ii) the density of males nm/A and (iii) the size of a small area surrounding the female where males are attracted to mate. This is thought of as a circular area with radius R. We remark that these assumptions are the same used in the theory of molecular collisions to deduce the law of mass action for gases of two diﬀerent types, with just the words “particle“ replacing “individual” and “type“ instead of “sex”. We write the relation between the system parameters in terms of some (unknown) function F ,

n c = F m , v, t, R . (2.1) A

By the Π-Theorem, see [1] or [25] for instance, equation (2.1) is equivalent to a relation that involves only the dimensionless quantities

vtRn vt Π = m and Π = , 1 A 2 R that is, vtRn vt c = f m , , (2.2) A R with f yet to be determined.

Next, we use data generated from agent-based simulations that count the number of contacts (with males) per female. For the simulations the individuals were programmed to follow the rules stated above and, for a single female, we counted a contact when its distance to a male is less than R. In the computations the units chosen for length and time were meters and hours (1hour = 1 time step). We used fixed values for the time 7 of observation, t = t∗ = 24(h), the radius R = R∗ = 0.05(m) and the number of males, nm∗ = 100, while varying area size A and individuals’ velocity v. On the plane v − A we arbitrarly chose the strip [25, 4 × 104] × [25, 2 × 103] and within this domain we fixed eight values of areas and twenty eight values of velocities. The values for the parameters were chosen in two blocks with the aim of capturing the surface characteristics at low and high parameter values. Then we produced 10 simulations for each corresponding pair of parameters and finally computed the contact averages. The simulations were run in NetLogo1 [30]. It is well known that the average number of contacts observed during a fixed period of time should increase with larger velocities and decrease with larger areas, therefore suggesting a relation of the form

avp c = , (2.3) Aq

where p, q and a are constants. This expression can then be rewritten in terms of the

dimensionless quantities Π1 and Π2,

K z }| { p−q q a p−q q vt vtRnm c = q p 2q−p Π2 Π1 = K . nm∗t∗R∗ R A

Using least squares to ﬁt relation (2.3) to the averages of the data points collected gave approximate values of p ≈ 1, q ≈ 1 and K = 1.2887. Denoting with nf the number of females and with C the total averaged number of encounters that females have with males, i.e. c = C/nf , gives

vtR C = K n n = KvtRAxy, (2.4) A m f

where x and y are the densities of males and females, respectively. The right hand term in (2.4) is traditionally obtained from the theory of molecular collisions, also known as the ideal gas model. That theory provides a constant value of K = 4/π = 1.2732..., which is in good agreement with the value obtained through our simulations (the relative error is less than 2%).

1https://ccl.northwestern.edu/netlogo/ 8

2.2.1 Pair formation We generate diﬀerent data by repeating the simulations of individual movement with the same assumptions as above but counting at most one mating encounter per female at each time step, i.e. discarding the possibility of simultaneous mating. Fitting the model to the new data produces p ≈ 1, q ≈ 1 and K ≈ 0.1231, which is less than 10% of the previous K value.

Our interest now is to include changes in movement direction at every step in time, depending on the previous direction rather than restricting individual movement to straight lines as in the assumptions. A limitation in the theoretical ideal gas model, as traditionally conceived, is that it does not capture the effects of the correlation in individuals’ movement on contact rates. From the dimensional analysis, however, we conclude that the degree of correlation should appear as a functional dependence between the value Kand the range of possible directions for individuals’ movement, i.e. K = K(θ), where at each time step each individual changes to a direction chosen uniformly at random from [−θ, θ] for a given θ, 0 ≤ θ ≤ π, independently of the other individuals. Repeating the numerical experiments but now allowing individuals to change movement direction with different degrees of correlation, we compute the average of K for different values of θ. Unsurprisingly, these K values appear almost constant, as first pointed out by Skellam [35] for the classical mass action, see Figure 2.1 (top). So it seems that K does not depend significantly on θ. This information can thus be used to approximate average mating encounter frequencies for individuals, given estimates of velocities, see Figure 2.1 (bottom). We emphasize, however, that the hypotheses made on the movement constitute a rough simplification of reality: males and females do not necessarly move at the same speed and the details of mating mechanisms have been deliberately left out.

To assess the variability due to movement correlation on the parameter K we re-ﬁt equation (2.3) with varying θ from 0 to π for the sets of points formed by averages ± (standard deviation) at each point of the v − A plane and obtained two new constants,

K+ and K−. The data suggested that for values of θ close to zero the diﬀerences are consistently larger, giving larger values for K+ and K−, see Figure 2.1. But with increasing θ, although the movement is more irregular, large variability in the values of

K+ and K− is absent. Our interpretation of this outcome is that, although individual movent is apparently more convoluted for larger θ, at each trial the (stochastic) process 9

Figure 2.1: Top: Averaged values of K obtained from observations where particles change their direction at each step in an angle chosen randomly from [−θ, θ]. The averages (stars joined with dash/dot curve) do not differ significantly from the mean K¯ = 0.1231. For each angle θ the circles mark the values of K+ and K−, which are the constants obtained by fitting the relation (2.3) to the original average data points ± standard deviation, respectively. Bottom: Mating encounter rates (encounters/h) for moving individuals as function of the population density (individuals/m2) and velocity (m/h). For illustration purposes, the detection radius R was fixed and chosen equal to 0.02 m and the velocities are in the range from 1 Km/h to 40 Km/h, which includes estimates for several insect species, see Table 2.1. 10

Table 2.1: Average velocities for sustained ﬂight in some insects, [28].

Insect Flight velocity (Km/h) Mayflies, small field grasshoppers 1.8 Bumblebees, rose chafers 3.0 Anopheles (malaria mosquitoes) 3.2 Stag beetle, damselfly, Ammophila (a fossorial wasp) 5.4 Housefly 6.4 Cockchafer, cabbage white butterfly, garden wasp 9.0 Blowfly 11.0 Desert locust 16.0 Hummingbird hawk moth 18.0 Honeybee, horsefly 22.4 Aeshna (a dragonfly), hornet 25.2 Anax (a dragonfly) 30.0 Deer botfly 40.0 is the same. On the contrary, for values of θ close to zero, the initial random directions for each individual define paths that look like the deterministic trajectories (θ = 0) in the domain (in this case, a torus). Those might differ completely each time the experiment is repeated because of the randomization of initial directions. This is independent of whether the boundary conditions are periodic or reflective.

2.3 Effects of environmental stochasticity The rate at which offspring are generated by a two-sex population in real scenarios is likely to be subject to random fluctuations due to environmental factors, like rainfall and temperature, but with sensitivity that is species-specific. If the law of mass action is chosen to model mating encounters, variability on the parameter K will appear as consequence of those fluctuations. This translates directly into variability of birth and death rates. As an illustration, let us initially consider the simplest deterministic population model with two sex ephemeral interactions,

1 x0 = −µx + P (x, y), (2.5) 2 1 y0 = −µy + P (x, y), (2.6) 2

where P (x, y) is the (symmetric) birth rate and µ is the death rate, which is assumed equal for both sexes. Suppose also that the initial sex ratio is 1:1. It is natural that we would like to use the law of mass action for P (x, y), i.e. proportional to xy, but as D.G. Kendall ﬁrst pointed out, this leads to solutions that blow up in ﬁnite time [21]. 11

One way this trouble can be fixed is by taking into consideration the average refractory time τ of females [2], during which a female avoids further sexual encounters just after successfully mating with a male. Let r denote the rate of mating encounters per female, r = KvRx = αx, then 1/r is the average time between encounters for one female. If the population densities are low enough so that 1/r is very much less than τ then the population growth will depend on the number of successful mating encounters made. In this case the average progeny produced by a female has to saturate when 1/r ↓ τ as the density of males increases. Therefore, the average birth rate per female is more 1 conveniently approximated with bp × αx × N+x , where b is the average offspring per female per encounter that survive to adulthood, p is the probability that an encounter produces offspring and N is the population density at which half of the females are able to reproduce. For simplicity we set N = 1, suggested by the reasonable approximation of the mean number of pairs formed at low densities, [15]. Given that initially we assumed a symmetric sex ratio, the whole population dynamics can be described by the equation

z2 γz (z − 2µ/γ) z0 = −µz + P (z/2, z/2) = −µz + bpα = , (2.7) 2(2 + z) 2 + z

where z = x + y and γ = (bpα − 2µ)/2 > 0. Thus, if z << 2 the population growth will approximately correspond to a mass action regime while if z >> 2 it grows exponentially. More precisely, z0 < 0 at the Allee threshold when z < 2µ/γ, giving rise to positive density dependence with a critical population size.

To integrate demographic random ﬂuctuations we only consider two types of events, birth and death of individuals, happening at exponentially distributed times with overall rate bpαz TR = + µ. 2(2 + z)

Thus, at each time step a death happens with probability q = µ/T R or a birth with probability 1 − q. In addition, we let K fluctuate randomly, with K ∼ Normal(K,¯ σ2) (K¯ = 0.1231) due to environmental changes and observe the effects for different values of σ2 through simulations. Figure 2.2 illustrates the cumulative probabilities of extinction up to time t, conditioned on extinction, with the initial population densities taken (i) at the Allee threshold value and (ii) half of it. The results suggest that an increasing variance in the random fluctuations associated to the mating encounters would induce earlier times of extinction. 12

Figure 2.2: Cumulative probability of extinction, conditioned on extinction up to time t = 2000 days. The effect of variability on the number of contacts, K ∼ Normal(K,¯ σ2), is shown for different values of σ2 = 0 (continuous line), 0.05 (dash- dots), 0.1 (dots). Top: The initial population value is taken equal to the Allee effect threshold 2µ/γ. Bottom: The initial population is half the Allee threshold. The simulations suggest that an increasing variance in the random fluctuations corresponding to mating encounters pushes the probability mass to the left, i.e. extinction is likely to happen sooner than expected. Parameters chosen for the simulations are 1/µ = 10(days), v = 1.33(Km/h), R = 0.02(m) and b = 3, in agreement with those characteristics in some relatively small insects. The value p = 0.01, which in practice depends on the complexity of the mating mechanisms, was set arbitrarly. 13

2.4 Discussion Mating processes in two sex insect populations are species-speciﬁc and generally involve vast complexity [3,8]. This makes it desirable to have an approximated description of the nonlinear process for mating encounters. The law of mass action has been traditionally used for this end even in cases where the mating mechanisms might be convoluted. The multiplicative constant for the mating encounter rate might be diﬃcult to estimate in many real applications. The novelty in this work is that, under general assumptions, we have approximated the value of that constant with the use of dimensional reduction and simulated data. Apparently, this constant has not been computed by analytic arguments.

The dimensional reduction approach also offers a way to examine the effects of correlation in the movement. The simulations suggest that in general, the variances of the values of K are small and similar when compared among results obtained with different degrees of correlation. Thus, the value K¯ can be used as the multiplicative constant in the expression for the encounter rate, KvRAxy¯ , independently of movement correlation. Relative larger variance values observed for highly correlated movement (θ close to zero) are attributed to the boundary conditions in the simulations. We remark that the program used to produce the data only counts ephemeral encounters and does not consider further association between paired individuals, as well as other complexities involved in mating mechanisms.

Small random fluctuations in the number of mating encounters can appear never- theless, caused by non-permanent random environmental changes. These effects might alter substantially the encounter rate (now random). As an illustration we simulated a basic stochastic birth/death process to obtain the probability distribution of the time to extinction, conditioned on extinction. The model has a deterministic skeleton for which extinction due to Allee effects is possible. The simulations suggest that small fluctuations on the parameter K¯ might induce relatively large increments on the (conditioned) cumulative probability of extinction, see Figure 2.2. For conservation or pest control efforts in which it is critical to assess how rapidly a population might become extinct, the approximation for K¯ presented here might help to shape more accurate quantitative predictions based on the law of mass action. 14

We would like to emphasize that, after all, mass action is a highly idealized model where physical and biological details are overlooked, including those corresponding to mating mechanisms. For instance, velocities associated with aerial mating in insects may be sensitive to the relative mass of flight muscle [12], while (the radius of) attraction may depend on chemical, visual or auditory signals [31, 33]. Trying to introduce these species-specific details into the mass action scheme could be challenging task that we have not considered. In the case that velocity and radius of attraction were both scaled to individual total mass, for example, it would be possible to insert these relations into the main formulae as long as the assumptions for the mass action still hold. If the case is that individual mass implies non-negligible (relative) size then the simulations cannot be used as we set them, and new computational experiments together with a re-assessment of the whole model have to be done. In this respect, we would like to highlight the interesting papers by E. Gurarie and O. Ovaskainen [16, 17] where, in contrast to the crude assumptions made for mass action, they present a refined theoretical framework that shows how encounter rates depend on the interplay of spatial distributions, scales of movement, individuals densities and inherent dynamics. 15

Chapter 3

A Model for Rat Management

The following is a paper in progress titled:

Induced infertility in rats: population dynamics and control

K. Snyder, B. Kohler, Luis F. Gordillo

A collaborative work with SenesTech

3.1 Introduction The use of sterilization techniques to control pest populations started with insects in the early 1950s and has a long history of success [22]. The enthusiasm produced by initial results in controling some species of insects reached a halt when scientists tried to extend the ideas to vertebrate populations. Despite the simplicity of the main idea behind sterilization, it is evident that difficulties due to technological constraints had to play a definite role at that time. The problems in implementing this kind of population control for vertebrate even go beyond those found in the sterilizing procedures for insects. For instance, the effects of the sterile individuals on population growth in mammals are expected to be proportional to the individual’s lifespan, especially if it overlaps several generations. In contrast, insect species on which the technique proved to be useful rarely live to see a new generation and, as consequence, continuous rearing and release of sterile insects is required [23, 24].

Recent developments in chemosterilant technology have proven efficient in producing infertility in rats. Animals feed on baits containing the chemicals and eventually their capacity to reproduce is significantly reduced and consequently so is the population density. Now, rat behavior depends on the size of the community to which it belongs. At low densities individuals show territorial behavior, i.e. they will reject individuals coming from other communities. Therefore, the main questions focus on finding under which circumstances low density levels can be reached and therefore induce territoriality. 16

In this chapter we present a conceptual model for rat populations subject to fertility control. As ﬁrst approximation, our intention is to produce qualitative predictions that focus on the possible modes of population behavior instead of exact numerical forecast. We present a general theoretical framework that, under reasonable biological assumptions, suggests when induced infertility might have chances of success.

We start by describing the dynamics of a spatially isolated rat population. The diﬀerential equations involved describe the transitions between the possible states that an individual may reach after baits with chemosterilant have been distributed: fertile, sterile and (for the case of females) gravid. Next we consider more than one rat community with individuals moving from one patch to another. Modeling movement between patches captures the territorial behavior of rats at varying densities and shows how the eﬀectiveness of the chemosterilant at reducing population density changes when interactions between rat communities occur.

3.2 Population dynamics in one patch Our initial step is to find in general terms an approximate description of the dynamics in a closed rat community (patch). We initially assume that (P1) males and females are homogeneously mixed and distributed in space, (P2) for males (females), sterile females (males) are indistinguishable from normal females (males), (P3) the average survival rate is the same for both genders, (P4) the mating system is polygynandrous, i.e. multiple males mate with multiple females, (P5) mating encounters are ephemeral, (P6) adult fertile females and males turn sterile by consuming the chemosterilant, which in average has the same effect on both genders, (P6) the chemosterilant has no effect on gravid females and juveniles (both genders), (P7) a mating encounter fails to produce offspring only when at least one of the individuals in the mating process has been sterilized.

3.2.1 Mating encounters Let φ(x, y) denote the rate at which mating encounters (pairs) are formed between x males and y available females. Only a fraction of these pairs will consist of sterile males and females. Any of the other three combinations will produce offspring. If we denote with F , Fs and Fr the densities of fertile, sterile and gravid females, respectively, and define M and Ms to be the densities of fertile and sterile males then the average fraction of pairs that will potentially produce offspring is given by χ = (MF + MsF + 17

MFs)/(M + Ms)(F + Fs). Thus, the rate at which mating encounters that produce oﬀspring happen is given by

Ψ = χφ(M + Ms,F + Fs).

The pairing function φ, also known as the mixing function, has several variations that have been extensively studied in demographic contexts [6]. For our computations we have chosen φ(x, y) = E min(x, y), where E is a constant that describes the percentage of mating encounters that will end up successfully producing oﬀspring.

3.2.2 Female and male sterilization The scheme in Figure 3.1 shows the expected dynamics in the female adult population, which in accordance to the initial assumptions, provides the following system of equations for transition rates between states,

  0  δλ  T (t) F (t) = −µF (t) + Ψ(tin) 1 − − Ψ(t) + (3.1) | {z } 2  K |{z} death | {z } | {z } new gravid new adults density dependance regulation −αF (t) + Fr(t − τr) − 1 − e F (t), | {z } recovery from | {z } refractory time fraction sterilized T (t) F 0(t) = 1 − e−αF (t) F (t) − µF (t) 1 − , (3.2) s s K T (t) F 0(t) = Ψ(t) − F (t − τ ) − µF (t) 1 − , (3.3) r r r r K where T = M + Ms + F + Fs + Fr is the total adult rat population and tin corresponds to the time point in the past when females that are currently producing oﬀspring had the corresponding mating encounter, i.e. became gravid,

tin = t − τa − τr . |{z} |{z} |{z} current time timespan to adulthood refractory time 18

Figure 3.1: Schematic diagram for the dynamics in the adult female population. The transition rates between states are given by the diﬀerential equations (3.1)-(3.3), where F , Fs and Fr stand for fertile, sterile and gravid females. The inﬂux of new adults corresponds to juveniles that reach maturity and death is regulated by natural causes only. Gravid females cannot turn sterile even if they eat the chemosterilant. Sterility is not reversible, even when rats stop eating baits with chemosterilant.

Table 3.1: Variables and parameters for the model

Variable Description Value Source F fertile female density - - Fs sterile female density - - Fr gravid female density - - M male density - - Ms sterile male density - - T total population density - - Parameter λ offspring per litter 6 - 10 - δ pup survival to adulthood - - K carrying capacity - - α “efficiency” of delivering the - - chemosterilant to rodents 1/µ adult time span less than 12 months [11] τa timespan to adulthood ≈ 3 months - τr refractory time after impregnation ≈ 23 days - E percentage of mating encounters that - - produce offspring

A complete list of parameters and their descriptions is given in Table 3.1. The corresponding rates describing state transitions in the male population are given by

δλ T (t) M 0(t) = −µM(t) + Ψ(t ) 1 − − (3.4) 2 in K − 1 − e−αM(t) M(t), T (t) M 0(t) = 1 − e−αM(t) M(t) − µM (t) 1 − . (3.5) s s K 19

Figure 3.2: Rat population density subject to sterilization in both sexes. Results obtained by solving numerically equations (3.1)-(3.5). Parameters used: δ = 0.5, E = 0.5, K = 1000, 1/µ = 5(months), (left) α = 0.01, (right) α = 0.1. Initial densities: (top) M = 300,F = 300,Fr = 150, (bottom) M = 8,F = 8,Fr = 4. Other values are chosen from the Table.

3.2.3 Numerical experiments The system (3.1)-(3.5) was solved numerically for fixed parameter values and the results plotted in Figure 3.2. For the computations it was assumed that only when a sterile female mates with a sterile male sterilization is effective, otherwise offspring is normally produced. Data from lab experiments show partial sterilization when one sterilized rodents mates with a non-sterilized. This issue will be addressed later with further study. For now we assume the worst case scenario.

We are also assuming complete efficiency of the chemosterilant given successful delivery, that is, rats that take on baits containing the chemosterilant will not produce offspring when mating with a partner in similar circumstances. They could, however, produce offspring if they mate with a partner that has not consumed the chemosterilant.

Over 50 simulations were done while varying α. It was seen that the parameter α, which denotes the eﬃciency of delivering the chemosterilant to rodents, plays a critical 20 role for determining the steady state of the system, as shown in Figure 3.2. The rat population (total) growth in time appears in Figure 3.2 for the values α = 0.01 (left) and α = 0.1 (right) with carrying capacity K = 1000. It seems that independently of the initial conditions, the population will stabilize around a stable population ≈ 600 for α = 0.01 and ≈ 200 for α = 0.1.

When the initial population is relatively high, close to the carrying capacity, sterilization will quickly change the growth trajectory (Figure 3.2, top panels). However, the same eﬀect is not seen in low density populations; in contrast with high rat population densities, low densities are harder to reach and reduction was observed in simulations only with higher eﬃciency on delivery.

3.3 Coupling patches

Let us consider now the case of two patches Π1 and Π2. First, we assume that the population in each patch follows the dynamics described by equations (3.1)-(3.5) if isolated. Second, let us allow movement of individuals between patches. The scheme in Figure 3.3 shows the expected dynamics in the female adult population with movement between patches. We add the assumptions that (P8) fertile and sterile individuals of both genders move between patches and may partake of food, water, and mate in either patch, (P9) gravid females and juveniles (both genders) remain at the nesting site and do not move between patches, (P10) bait with chemosterilant is only available at patch

Π1, (P11) males (females) may move at diﬀerent rates from Π1 to Π2 than from Π2 to

Π1, (P12) rats are most territorial at some given population size, Tˆ, at which no rats are permitted to enter in from other patches. Equations for this model are expanded from those for the one patch model.

3.3.1 Movement between patches Flow of individuals between patches is dependent upon rat territorial behavior. Territoriality is assumed to be such that (i) populations too large or too small will be unable to keep out neighboring rats, (ii) at a particular density, Tˆ, rats prevent individuals from neighboring patches from entering their patch. This movement is modeled with the ﬂow of individuals proportional to a symmetric sinusoidal curve with a minimum at the density Tˆ, ρy(x − Tˆ)2/(1 + (x − Tˆ)2), 21

Figure 3.3: Schematic diagram for the dynamics in an adult female population with an interactive model. The transition rates between states are given by the diﬀerential equations (3.14)-(3.23). The chemosterilent is only present in patch Π1. All adult female categories are also subject to deaths by natural causes. The inﬂux of new adults corresponds to juveniles that reach maturity. Gravid females and juveniles remain in their current patch until they return to the fertile adult stage. Gravid females cannot turn sterile even if they eat the chemosterilant. Sterility is not reversible, even when rats stop eating baits with chemosterilant. where x is the total population in the patch, y is the density of individuals of a particular class (e.g. sterile females) that is moving to another patch and ρ is a positive constant that represents the average rate at which individuals move to another patch in absence of any constraint. In other words, 1/ρ is the average time that an individual spends at a patch. This value might depend on factors that include variable environmental conditions, but we assume here that it is constant.

With this we deﬁne movement for diﬀerent classes of individuals in our context of two patches in equations. For instance

ˆ 2 0 dF12 (T2 − T2) F = = ρ1F1 12 2 dt 1 + (T2 − Tˆ2) represents the number of fertile females moving from patch Π1 to Π2 per month where

F1 is the number of fertile females in patch Π1 and T2 is the total population of rats in patch Π2. For sterile groups

ˆ 2 0 dMs21 (T1 − T1) M = = γ2Ms2 s21 2 dt 1 + (T1 − Tˆ1)

represents the number of sterile males moving from patch Π2 to patch Π1 per month 22 where Ms2 is the amount of sterile males in patch Π2 and T1 is the total population of rats in patch Π1 with the other expressions deﬁned similarly. In Figure 3.4 we see that

flow of fertile females into patch Π1 is greatest when the total population Π1 is either small or large meaning territoriality is not effective. However, when the population is close to Tˆ1, territoriality becomes more effective and less movement into patch Π1 occurs.

When the total population is equal to Tˆ1 the number of individuals entering is zero. The rates of movement are deﬁned by the following

2 2 0 (T2−Tˆ2) 0 (T2−Tˆ2) F12 = ρ1F1 2 (3.6) M12 = γ1M1 2 (3.10) 1+(T2−Tˆ2) 1+(T2−Tˆ2) 2 2 0 (T1−Tˆ1) 0 (T1−Tˆ1) F21 = ρ2F2 2 (3.7) M21 = γ2M2 2 (3.11) 1+(T1−Tˆ1) 1+(T1−Tˆ1) 2 2 0 (T2−Tˆ2) 0 (T2−Tˆ2) Fs12 = ρ1Fs1 2 (3.8) Ms12 = γ1Ms1 2 (3.12) 1+(T2−Tˆ2) 1+(T2−Tˆ2) 2 2 0 (T1−Tˆ1) 0 (T1−Tˆ1) Fs21 = ρ2Fs2 2 (3.9) Ms21 = γ2Ms2 2 . (3.13) 1+(T1−Tˆ1) 1+(T1−Tˆ1)

Figure 3.4: Rate of fertile females movement from Π2 to Π1 (equation (3.7)) as function of population density of Π1. It is assumed that rats will show territorial behavior at a density Tˆ, moving the least when the density is furthest from Tˆ whether smaller or larger. The parameters used were chosen arbitrarly: Tˆ1 = Tˆ2 = 10, K1 = K2 = 20, ρ1 = ρ2 = 4, γ1 = γ2 = 4. 23

3.3.2 Territoriality A reason for introducing patches is to include the effects of rat territorial behavior on population dynamics. In an interactive environment, if all adult rats were removed from a specific area it will quickly be repopulated by adults coming from neighboring communities and the population will grow back to full density [34]. Territoriality then might be the key to maintaining a manageable population in a specific area. Rat territorial behavior is mainly determined by female nesting sites; daily movements however are related instead to resources available [34]. Thus our model assumes that rats move between patches for food, water, and mating interactions but while a female is gravid she will remain stationary, at the nesting site. For management purposes, we assume there is an ideal population in which the rats maintain a manageable low density yet are large enough to keep neighboring rats from infiltrating. When this assumed ideal population is not reached the population size of the patch no longer reflects the number of rats being managed within the patch, i.e. only the rats currently in the patch contribute to the measured population size but rats in outside patches also frequent the patch adding an unmarked number of rats to the population being managed.

To model this type of behavior we used a symmetric sinusoidal function to model the rate of individuals that move between patches. This function reaches its minimum at an ideal population Tˆ at which no rats from outside patches may enter (maximal territoriallity). Otherwise, when the population is above or below Tˆ territoriality is less eﬃcient and movement into the patch increases, adding an unknown number of rats to the population being managed. The equations for patch Π1 are deﬁned as follows

   δλ  0  0 0  F1(t) = −µF1(t) + Ψ(tin) + F21(t) − Fs12(t)  · (3.14)  2 | {z } | {z }  fertile female sterile female movement from movement from Π2 to Π1 Π1 to Π2 T (t) 1 −αF1(t) · 1 − − Ψ(t) + F (r(t − τr) − 1 − e F1(t), K1 0 −αF1(t) 0 0 Fs1(t) = 1 − e F1(t) + −µFs1(t) + Fs21(t) − Fs12(t) · (3.15) T (t) · 1 − 1 , K1 0 T1(t) Fr1(t) = Ψ(t) − Fr1(t − τr) − µFr1(t) 1 − , (3.16) K1 24

   δλ  0  0 0  M1(t) = −µM1(t) + Ψ(tin) + M21(t) − M12(t)  · (3.17)  2 | {z } | {z }  fertile male fertile male movement from movement from Π2 to Π1 Π1 to Π2 T (t) 1 −αM1(t) · 1 − − 1 − e M1(t), K1 0 −αM1(t) Ms1(t) = 1 − e M1(t) + (3.18) 0 0 T1(t) + −µMs1(t) + Ms21(t) − Ms12(t) 1 − . K1

Similarly the corresponding equations for patch Π2 are

δλ F 0(t) = −µF (t) + Ψ(t ) + F 0 (t) − F 0 (t) · (3.19) 2 2 2 in 12 21 T (t) · 1 − 2 − Ψ(t) + F (t − τ ), K − 2 r r 0 0 0 T2(t) Fs2(t) = −µFs2(t) + Fs12(t) − Fs21(t) 1 − , (3.20) K2 0 T2(t) Fr2(t) = Ψ(t) − Fr2(t − τr) − µFr2(t) 1 − , (3.21) K2 δλ M 0 (t) = −µM (t) + Ψ(t ) + M 0 (t) − M 0 (t) · (3.22) 2 2 2 in 12 21 T (t) · 1 − 2 , K2 0 0 0 T2(t) Ms2(t) = −µMs2(t) + Ms12(t) − Ms21(t) 1 − (3.23) K2 with φ(x, y) and Ψ deﬁned as in equations (3.1)-(3.5).

3.3.3 Numerical experiments The system (3.14)-(3.23) was solved numerically for ﬁxed parameter values and cases for which the population Π1 stabilizes at Tˆ within 5 and 10 years were determined.

Figure 3.5 shows cases for which stabilization occurs at Tˆ in patch Π1 varying with regard to time spent in each patch. It is shown that when rats spend more time in a patch (1 month), the likelihood of stability at Tˆ is relatively low when compared to shorter time spent in a patch (1-2 weeks). Cases where the population fails to stabilize at Tˆ are considered less desirable as there will continue to be problems with incoming rats. We also are only looking at the stability dynamics in patch Π1 as this is the patch where the chemosterilant is released and thus assumed as the patch of interest.

Figure 3.6 shows an example of a population which stabilizes at Tˆ (bottom) and 25

Table 3.2: Variables and parameters for interactive model

Variable Description Value Source Fx fertile female density from patch x - - Fsx sterile female density from patch x - - Frx gravid female density from patch x - - Mx fertile male density from patch x - - Msx sterile male density from patch x - - total population density from T - - x patch x Parameter λ offspring per litter 6-10 - δ pup survival to adulthood - - Kx carrying capacity for patch x - - “efficiency” of delivering the α - - chomosterilant to rodents 1/µ adult time span less than 12 months [2] τa timespan to adulthood ≈ 3 months - τr refractory time after impregnation ≈ 23 days - percentage of mating encounters E - - that produce offspring average time (months) females 1/ρ - - x spend in patch x average time (months) males 1/γ - - x spend in patch x total population of patch x at Tˆ - - x which no rats from other patches are allowed in (maximal territorial)

one that does not (top). When Tˆ is reached movement into patch Π1 ends and the population becomes constant. In our numerical solutions we used a single value of Tˆ = 500. The actual value for Tˆ is yet to be determined as well as whether it changes with respect to the density of the population or the area in which it is located. Once Tˆ has been determined numerical solutions can be repeated for the given Tˆ to determine initial conditions for which stabilization at Tˆ is likely to occur.

3.4 Using GIS to determine patches Modeling with patches requires an understanding of how rats disperse in space for which we suggest using Geographic Information Systems (GIS). GIS is a system used to store and manage spatial or geographical data. GIS have been widely used in the past to help map the distribution of animal and pest populations [13]. GIS was used by Feldmann and Ready to model the distribution of genetic populations of the tsetse and screwworm ﬂies and by Roberston, Simmons, Jarvis, and Brown together with data from the Southern African Bird Atlas Project (SABAP), where reports of various bird species in South Africa were recorded for six years to determine the distribution, to 26

Figure 3.5: Stability at Tˆ (red dots) within 5 years (left) and 10 years (right) occur when maximal territorial behavior of rats in patch Π1 prevent rats from outside patches to enter. Densities which do not stabilize at Tˆ (blue triangles) continue to have movement into patch Π1 from outside patches. Failure to stabilize within 10 years does not mean the populations will never stabilize at Tˆ. Graphs vary with regard to the time 1 1 spent within a patch - (time females spend in Π1), (time females spend in Π2), ρ1 ρ2 1 1 (time males spend in Π1), (time males spend in Π2). Top: ρ1, ρ2, γ1, γ2 = 1 γ1 γ2 (1 month), center: ρ1, ρ2, γ1, γ2 = 2 (approximately 2 weeks), bottom: ρ1, ρ2, γ1, γ2 = 4 (approximately 1 week). When rats spend less time in a patch (1-2 weeks) more populations stabilize at Tˆ than in populations where rats spend more time (1 month) in a patch. Results obtained by solving numerically equations (3.14)-(3.23). Thirty initial populations were chosen at random for Π1 and Π2. For each initial population half are male and half female, of the females two thirds start as fertile and one third start as gravid. Other values are chosen from Table 3.2. Parameters used: δ = 0.5, E = 0.5, 1/µ = 5(months), Tˆ = 500, K1 = 800, K2 = 1600, α = 0.3. 27

Figure 3.6: Rat population density subject to sterilization in patch Π1. Results obtained by solving numerically equations (3.14)-(3.23). Each plot shows the change in rat population densities (patch Π1 top, Π2 bottom in each figure) for the total population (right) and separated by type (left): male, female, sterile female, gravid female, and sterile male. The bottom figure is an example of stabilization in patch Π1 at the population Tˆ = 500, where territorial behavior is maximized and no rats from outside patches are permitted to enter. The top figure is an example of stabilization above Tˆ, around 680, where movement between patches continues. Parameters used: 1 µ = 5(months), δ = 0.5, λ = 8, α = 0.3, E = 0.5, ρ1 = ρ2 = 2, γ1 = γ2 = 2, K1 = 800, K2 = 1600, and Tˆ = 500. Initial densities: (Top) P opulation1 = 500, P opulation2 = 111, (Bottom) P opulation1 = 697, P opulation2 = 221. For each initial population used half are male and half female, of the females two thirds start as fertile and one third start as gravid. Other values are chosen from Table 3.2. 28 approximate absolute abundance for use in estimating population size [13][32]. For our work we recommend the use of open data for 311 service requests that record the location, date and type of service request made throughout a city. Filtering the list to only include service requests involving rodents we can then use the information to estimate the rat densities by location.

Figure 3.7 shows an example of GIS to estimate rat densities. The ﬁgure is a basemap of New York City and surrounding areas overlayed with a raster image. A raster image is a matrix of pixels organized on a grid where each cell contains a value representing information. In this case the pixel values represent an estimated population density for rats in NYC in 2009. 311 service requests for NYC in 2009 were used to identify rodent related service requests then we estimated an expected number of rats per request and plotted the data [9]. The rat densities can then be seen, low densities in blue and high densities marked in red. These estimates can be made for all cities with available 311 service request data, including NYC, NY; Chicago, IL; San Francisco, CA; and Boston, MA. This information can then be used to estimate patch sizes and with the mathematical model, determine where the chemosterilent will be most eﬀective in reducing the rat population.

3.5 Conclusions Induced sterility in insects has been practiced and mathematically modeled but the same has not been done for vertebrates, whose biology and behaviors diﬀer greatly. With new technologies producing infertility in rats there is now a need for conceptual models that help to clarify how the population dynamics depend upon the control and environmental parameters.

We have presented two models for invoked sterility in rat populations. Our first model assumes a closed rat community where we see the effects of sterility in both males and females as well as its overall effect in reducing the population. From the numerical simulations it becomes apparent that the effectiveness of delivering the chemosterilant to rodents is critical in reducing the population. Also, the size of the initial population for which the chemosterilant is introduced changes the growth of the population: when the initial population is close to the carrying capacity a prompt reduction may be seen, but damped oscillations are likely to occur before stabilizing. With a relatively low initial population an increase may be seen initially before stability occurs. The initial 29

Figure 3.7: Raster image of estimated rat densities in NYC 2009. The data comes from NYC 311 service request data in 2009 related to rodents. For each type service request an expected number of rats was assigned and the sum of expected numbers is taken at each location plotted as a dot on the image. Low expectancies are plotted in blue (min = 0) up to high expectancies plotted in red (max = 117). The expected numbers used for each type is as follows: E(Conditions Attracting Rodents) = 1, E(Signs of Rodents) = 2, E(Rat Sighting) = 3, E(Mouse Sighting) = 0, and E(Rodent Bite - PCS Only) = 1. population does not seem to greatly impact where the population stabilizes, only its growth to stability. The biological interpretation is that at low densities, it is harder to reach rats with the chemosterilant.

The second model looks at connected rat communities where movement of individuals between patches and territorial behavior in each patch occur. Assuming there is a low rat density Tˆ at which territorial behavior is highest (no rats from outside patches are permitted in) we have modeled the effects of sterility within a given patch while movement between patches occur. While Tˆ is currently unknown, the reduction of a population to Tˆ is desirable as a manageably low population that self-prevents infesta- tion from surrounding patches. Again, the efficiency at delivering the chemosterilant is critical in reducing the population to Tˆ as is the initial population. However which initial populations are likely to stabilize at Tˆ is dependent upon the carrying capacities of the focal patch and its surrounding patches as well as the average time spent in a patch. When less time is spent in a patch there are more initial populations that stabilize at Tˆ than do when longer time is spent within a patch. The initial populations that do 30 result in stability at Tˆ depend on the parameters used but it is often seen that low inital populations are more likely to stabilize at Tˆ. Thus effectiveness in population control is likely to increase with

• Increased eﬀectiveness of delivering chemosterilant

• Decreased average time that an individual spends within a patch

• Decreased initial population for which the chemosterilant is introduced.

Some of the biological parameters for our models (see Tables 3.1 and 3.2) remain unknown and further studies of rat behavior may need to be done to determine their values. For instance, the carrying capacities of a patch, average time spent within a patch for males and females, and Tˆ for which maximal territoriality occur need to be estimated for our model to be most productive. For these to be estimated the aﬀected patches must also be determined. One possibility for estimating patches (and their initial population densities) is through a city’s 311 service reports and a GIS program we have developed to see where rat activity is reported within a city and estimate the population density from the quantity of reports. While this method can be used for cities with 311 service request data, other methods will be needed for cities without this information.

Other factors that may change the outcome of invoked sterility which are not accounted for in our models include

• Reduced competitivity due to hormonal changes in both males and females,

• Eﬃciency in measuring wild population densities,

• Role of human environment: urban vs rural and geographical locations (in terms of cultural practices), and

• Change in behavior due to seasonal ﬂuctuations.

As these and other factors are studied it may be necessary to change the model to account for those factors that are most influential in modeling the effects of sterility. The models also may need to be adapted if complete efficiency of the chemosterilant is not guaranteed and rats may change back from sterile to fertile. Likewise the pairing function used can change if sterility occurs differently than described for these models. 31

The models provided here are the start in modeling induced sterility in vertebrates and can be built upon and adapted for other vertebrates as well as improved modeling of rodents. 32

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APPENDICES 37 38

Appendix A

Pair Formation Data

Table A.1: Average number of encounters for given areas and velocities when θ = 0, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 13.1 12.4 7 1 1.2 0.5 0.3 0.4 50 25.4 23.9 17 2.7 2.2 1.1 0.8 0.3 75 38 40.1 23.8 4.2 1.9 1.7 1.1 0.2 100 49.6 51.9 30.1 5.5 2.6 1.9 1.5 0.9 200 104.5 105.4 58.8 8.6 6.4 3.2 3.4 2.2 300 163 152.6 91.9 12 9.3 5.7 3.8 2.5 400 215.6 209.4 132.3 16.8 9.5 9.1 5.9 4 500 253.5 260.3 149.4 22.1 15.8 10.3 9.1 5.2 600 309.5 304.9 174.1 27.3 16.3 12.3 8.6 7.5 700 360.1 356.6 198 28.8 21.1 13.6 12.4 5 800 410.9 413.1 235.9 32.3 25.2 15.3 12.4 8.9 900 459.5 509.1 280.1 37.6 24.8 18.9 15.1 8.7 1000 508.1 515.8 289.5 42.9 25.5 20.8 15.8 9.8 1200 607.7 618 360.8 52.7 32.9 23 19 10.3 Velocity 1400 714.6 709.3 427.5 62.6 36.8 30.9 23.8 14.4 1600 825.6 808.9 462.6 68.2 45.2 31.8 22.6 14.9 1800 913.5 920.9 521.7 75.4 50.4 34.5 29.3 15.8 3000 1650.8 1543 864.9 126.6 83.7 55.8 48.7 25.1 3200 1625.3 1624.2 908.4 137.6 90.2 58.8 50.6 28.9 5400 2754.5 2754.2 1552 223.8 147.2 100.5 77.8 52.4 6400 3197.6 3416.2 2022.8 272.9 177.3 119.4 102.9 76.5 9000 4595.6 4766.9 2775.3 381.9 242 176.8 135.9 83.9 11000 5636.3 5612.1 3158.5 460.9 304.8 214.5 174 102.8 16000 8960.6 8140.9 4599.3 675.4 444.8 305.8 233.8 146.9 18000 9158.8 9206.1 5202.3 768.8 489.6 345.8 275.4 162.4 22400 11457.5 11470.5 6464.4 1042.1 618.6 426.5 339.2 203.3 25200 12779 12843.4 7298.1 1056.1 926.5 493.9 369 227.7 40000 20700.7 22027.2 11586.8 1674.7 1087.1 761.2 591 368 39

Table A.2: Average standard deviation of number of encounters for given areas and velocities when θ = 0, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 2.5865 4.9031 1.7321 0.7746 1.249 0.6708 0.4583 0.6633 50 4.0299 5.5218 4.8166 1.4177 1.4 1.1358 0.6 0.4583 75 7.7846 12.5096 4.6861 2.8213 1.0440 1.1 0.8307 0.4 100 4.0299 7.8797 6.3159 3.1701 1.6852 1.3748 1.0247 1.0440 200 12.532 25.6289 4.7074 2.8 2.6153 2.1817 2.0591 1.3266 300 12.7515 13.7928 22.0248 3 3.6069 1.4866 1.6613 1.6882 400 21.2612 15.1539 33.1634 2.4 3.0083 4.3232 3.048 1 500 16.3783 16.7574 12.5156 4.0361 3.4871 2.3259 1.4457 1.99 600 11.4302 20.2457 12.965 7.7595 2.7946 3.0348 2.3324 5.8352 700 15.4107 26.7589 13.1985 2.9597 4.1821 3.1686 3.0397 1.8439 800 26.5761 19.6441 16.4162 7.1141 10.2548 3.0348 3.0725 2.5865 900 15.6157 100.3877 62.8641 8.1265 4.8539 3.9611 3.0806 2.4104 1000 16.8846 93.5284 11.758 6.6551 3.4132 4.2615 2.2716 3.763 Velocity 1200 27.0815 28.6810 49.5839 5.9 4.1821 4.1952 3.821 3.2573 1400 30.0905 29.0656 71.1607 7.4592 5.4553 6.3632 4.2143 3.7736 1600 15.3571 28.9878 20.4705 7.1106 7.2498 2.6758 4.5431 4.3232 1800 29.9742 25.205 16.5230 6.4062 5.9025 4.4777 4.9 4.7707 3000 324.2807 48.4912 35.5625 8.1511 8.3791 2.9597 5.7454 4.989 3200 35.1171 29.0475 26.2876 7.6837 9.9177 6.9971 4.9437 5.8215 5400 61.3404 62.3695 29.4415 16.9104 14.0769 5.8523 8.0598 6.9886 6400 71.1143 523.0133 576.7214 10.2708 15.8874 13.8723 6.5031 58.2482 9000 83.6973 514.1211 493.2298 19.0759 23.2981 8.7955 8.8255 12.1856 11000 156.6640 86.2699 72.4779 26.4479 25.4039 12.0354 13.6528 9.7139 16000 1799.7636 70.435 49.6247 15.9261 25.5648 11.6516 16.2960 14.5221 18000 101.4296 106.4936 80.6028 20.6436 23.6821 16.1914 16.2616 10.8554 22400 176.183 121.9617 124.1315 287.976 33.7674 15.7242 15.2696 15.8054 25200 235.783 228.6846 105.5277 28.5778 692.3043 22.4698 14.9198 17.7429 40000 853.8469 4380.3194 164.7032 55.0255 35.9373 32.3784 22.2171 16.0187 40

Table A.3: Average number of encounters for given areas and velocities when θ = 90, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 11 11.9 9.7 0.7 1 0.5 0.1 0.4 50 20.9 26.7 15 2.4 1.4 1.7 0.7 0.5 75 37.1 38 22 3.6 1.6 1.6 1.3 0.5 100 50.2 51.3 28 3.9 2.4 1.3 2.2 0.9 200 106.8 104 58.8 8.5 5.6 4.2 3.4 1.2 300 155.9 155.2 87.7 13.4 8.7 5.5 3.8 2.6 400 213.4 208.3 112 16.7 11.5 6.4 5 3.2 500 254.7 256.7 144.9 23.1 14 8.4 8.2 3.8 600 315.7 300 179.9 22.8 19.4 10.7 8.2 5.5 700 374.6 365 209.1 29.8 20.9 16.9 11.3 6.8 800 410.2 408.9 238.4 35.4 21.1 20.2 12.5 7.4 900 461.2 446.9 261.7 40.9 26.1 16.8 14.2 8 1000 521.7 515 291.4 47 28.8 19.8 13.1 9.7 1200 624.8 622.3 354.7 49.3 35.5 21.5 17.7 12.1 Velocity 1400 731.8 724.9 409.9 58.8 37.6 20.9 13.7 13.7 1600 811.9 829.3 462.2 65.3 45.9 30.9 26.1 15.9 1800 932.5 933.4 522.3 76.6 49.5 33.6 30.9 15.2 3000 1545.6 1549 871 120.5 84.5 57 46.9 26.7 3200 1642 1652.8 943.7 136.9 94.3 62.3 47.5 29.9 5400 2791.1 2767.9 1577.6 233.3 146.6 110.6 80.9 47 6400 3307.4 3311.1 1875.8 276.4 183.2 124.2 96.8 54.3 9000 4658.4 4682.8 2659.3 378.4 249.9 172.4 138.9 82.9 11000 5691.3 5707.6 3247.5 473.5 311.1 213.4 168.1 100.4 16000 8245.3 8335.5 4665.5 678.9 450.3 315.3 243.2 152.6 18000 9344 9296.2 5252.1 761.9 512.3 360.7 281.7 163.3 22400 11591.7 11602.2 6587.5 974 433.9 621.1 209.9 343.2 25200 13083.4 13025.2 7340 1076 701.6 489 384.5 229 40000 20743.3 20668.4 11694.4 1704.8 1115.2 784.9 615 377.6 41

Table A.4: Average standard deviation of number of encounters for given areas and velocities when θ = 90, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 4.0743 3.4482 4.0755 0.6403 0.8944 0.6708 0.3 0.4899 50 3.8328 5.9338 3.7683 1.6852 0.4899 0.4583 0.6403 0.922 75 4.3692 6.0663 4.9598 1.562 1.1136 1.3565 0.9 0.6708 100 5.9296 5.3675 4.7117 1.9723 2.1541 1.7349 1.7205 1.044 200 10.2059 9.0333 9.2715 2.5397 2.4166 1.8868 1.7436 0.8718 300 11.0313 10.4766 11.4896 3.8781 2.6851 1.8574 1.6 1.2 400 15.97 11.3846 6.7823 4.6701 3.9306 2.1071 1.6125 1.833 500 16.162 10.3832 7.7389 3.048 2.4083 3.6932 2.5219 1.7205 600 16.9885 10.7051 13.2322 3.0919 5.3703 2.9682 3.0594 2.0616 700 12.7765 10.9727 17.2305 4.2143 5.0289 2.0224 3.2265 3.1241 800 14.3861 17.1898 13.6543 4.2474 4.134 5.1342 3.9051 3.2 900 20.9227 18.8491 9.1657 5.1468 4.3692 5.4736 3.6551 2.6077 1000 18.2321 12.1078 18.0566 7.0143 4.9759 3.6824 3.0806 2.6096 Velocity 1200 24.3261 29.4416 15.818 6.1976 4.801 5.4083 3.743 2.9138 1400 14.5245 15.9653 23.4327 12.9213 5.004 4.826 3.2265 3.2265 1600 28.8772 34.8082 12.5443 7.6948 9.3536 5.009 5.4672 3.8846 1800 35.8281 34.7396 20.4551 4.8208 3.5 4.9639 5.1856 3.0919 3000 35.2483 39.512 28.7193 12.7613 7.311 7.0993 9.4387 5.6223 3200 40.7529 42.1706 31.4167 12.8721 6.5734 9.5609 7.1169 4.2532 5400 40.9865 54.9026 37.7444 15.8748 7.3919 16.2862 6.2201 6.4031 6400 49.506 53.7019 21.7936 12.0183 16.0736 10.6 9.988 7.3355 9000 58.3184 89.5241 39.2455 23.1396 11.6486 11.4035 12.2593 7.6085 11000 81.593 65.6433 50.8336 12.9634 9.0934 8.2122 10.0941 9.0796 16000 72.7435 58.6502 62.2676 20.3 16.9059 22.9741 17.2673 13.5956 18000 76.5585 85.593 48.3021 24.1183 18.4068 21.7028 20.7463 10.3542 22400 78.73 94.4784 111.3061 34.6208 22.5009 16.8134 16.8728 16.4791 25200 120.534 118.5275 62.2447 38.8021 20.9962 20.9952 11.5087 15.582 40000 133.2344 138.6479 104.2240 27.0695 31.8302 20.0372 26.7881 14.2281 42

Table A.5: Average number of encounters for given areas and velocities when θ = 180, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 14.9 13.9 7.6 1.2 1.1 0.4 0.2 0.3 50 27.9 23.8 16.6 2 0.9 0.9 0.5 0.4 75 39.5 40.7 21.6 2.8 1.5 1.4 0.9 0.4 100 52.3 50.9 30.6 5.8 2.5 2.9 1.6 1.1 200 103.5 102.8 57.6 8.4 3.8 4.1 3.1 1.5 300 149.6 155.5 88.5 12.4 6.6 5.7 3.8 2.8 400 209.4 215.9 110.8 17.1 12.4 7.6 5 2.8 500 258.2 260 150.4 23.4 13.9 9.5 7.3 5 600 314.2 315 174.7 23.7 15.6 12 10 4.9 700 364.4 357 207 29.4 21.1 12.7 10.9 6.8 800 420.4 422.5 229.5 34 22.7 13.4 11.7 7.3 900 460.7 463.7 258.9 39.6 25.5 17.2 15.1 7.8 1000 516.3 515.8 293.1 43.6 28.4 20.1 14.1 10 1200 628.2 638.3 352.1 56.9 35.6 23.9 20.5 10.8 Velocity 1400 712.6 738.4 410.3 62.8 39.2 29.8 22.1 15.4 1600 819.7 824.1 480.4 68.5 42.5 33.2 24 14.1 1800 935.1 919.9 535.3 79.3 50.6 36.2 23.7 16.4 3000 1568.3 1562.1 879.9 129.4 85.4 60 43.8 27.2 3200 1653.4 1659.7 944.6 137.3 91 62.3 51 27 5400 2802.4 2784.8 1576.6 232 142.7 106.9 88.2 50.8 6400 3302.9 3332 1897.3 279.7 180.2 125 100.3 60.5 9000 4654.8 4684.1 2639.1 386.4 245.8 177.8 141.3 81.4 11000 5686.1 5697.5 3230.4 464.3 308.9 216.1 170.2 100.8 16000 8276.1 8278.6 4688.6 688.8 439.4 307.9 236.6 152.5 18000 9361.1 9388.5 5262.4 773.4 492.3 360.7 273.9 169.4 22400 11635.8 11597.4 6581.9 613 623.1 432.2 348.6 207.7 25200 13086.4 13054.1 7347.2 1081.7 705.6 491.2 400.8 232.3 40000 20725.4 20624.4 11658.3 1722.4 1119.6 785 601 371.4 43

Table A.6: Average standard deviation of number of encounters for given areas and velocities when θ = 180, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 4.08534 3.72693 1.11355 0.87178 1.37477 0.66332 0.4 0.45826 50 5.08822 3.76298 2.8 0.89443 0.83066 0.9434 0.67082 0.4899 75 4.98498 7.65572 2.90517 1.53623 1.20416 1.28062 1.13578 0.4899 100 6.05062 5.46717 5.55338 2.89137 2.06155 1.04403 1.28062 0.9434 200 7.53989 10.47664 6.03656 2.53772 2.22711 1.64012 1.13578 1.11803 300 10.79074 15.6285 7.77496 3.07246 2.00998 1.79165 1.98997 1.46969 400 10.99272 8.22739 9.73447 3.23883 3.03974 1.95959 2.14476 1.16619 500 16.06113 8.46168 9.48894 2.6533 3.93573 3.69459 3.28786 1.73205 600 17.29624 13.25142 11.51564 4.87955 3.66606 3.34664 1.84391 2.42693 700 13.66894 13.82751 10.77961 5.49909 3.47707 3.06757 2.21133 1.8868 800 19.40722 9.33006 20.8866 3.2249 5.81464 4.02989 2.45153 1.95192 900 15.35611 21.56409 14.72719 3.8 4.5 6.85274 3.26956 2.08806 1000 27.26921 25.61952 16.64001 6.74092 6.32772 4.7 2.77308 3.40588 Velocity 1200 24.68522 20.55262 20.9497 6.5643 2.83549 5.43047 4.15331 1.72047 1400 25.80775 24.2619 23.77415 10.59056 6.2578 3.45832 5.04876 3.49857 1600 25.54623 27.52617 26.27242 3.77492 4.56618 6.9685 5.56776 4.03609 1800 29.70673 25.99788 24.25304 7.88733 8.96883 6.02993 5.29245 2.90517 3000 47.7306 35.12108 23.4284 12.62696 10.02198 7.02851 6.67533 4.42267 3200 36.269 37.17271 29.86369 6.78307 8.34266 7.26705 3.89872 7.28011 5400 83.55142 46.30292 33.32327 17.05286 11.25211 7.3 13.25745 5.05569 6400 68.03154 55.52657 49.06536 26.43501 14.5451 12.83745 10.07025 9.54201 9000 82.28584 65.24485 57.19694 13.75645 13.33267 11.9063 10.70561 8.55804 11000 78.58047 68.02242 39.96298 19.17837 20.91148 12.755 12.52837 13.33267 16000 96.68656 99.34304 43.98909 21.78899 22.43301 17.66041 17.11257 12.95569 18000 101.9975 82.70339 67.49252 35.58146 17.59005 15.78005 17.30000 13.74191 22400 80.96271 105.9077 70.19038 38.6057 30.43173 17.41149 19.04836 16.99441 25200 149.93945133.99735 64.66962 22.68061 27.99357 18.52998 16.33279 12.82225 40000 114.50869164.33454 89.93448 40.63791 28.99379 26.70206 30.99355 17.69294 44

Table A.7: Average number of encounters for given areas and velocities when θ = 270, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 13.2 13.5 7.4 1.4 0.5 0.7 0.3 0.3 50 27.3 26.3 13.6 2.6 1.6 0.6 0.7 0.2 75 41 35.6 19.8 3.6 1.6 1.8 1.1 0.5 100 49.7 49.1 28.2 4.9 2.4 2.6 2.1 0.6 200 96.1 102.2 53.8 27.9 6.6 3.9 3.7 1.1 300 149.6 149.2 81.5 11.9 8.6 6.4 3.7 1.9 400 194 199.1 111.6 17.1 10.3 9.3 6.6 2.7 500 256.3 255.2 142.1 20.6 12.7 8.9 8.4 4.4 600 299.3 296.4 172.5 11.1 26.9 16.7 9.9 6.4 700 343.4 353.9 195.5 27.8 19.1 15.3 11.1 6.4 800 427.2 429.3 234.2 33.8 20.2 15.4 11.8 7.6 900 468.5 470 259.6 42.6 26.1 18.6 14.2 8 1000 518.3 525.7 288.3 41.4 28.1 20.4 14.9 9.6 1200 617.8 623.4 356.1 53.8 32 23.5 17.6 11.7 Velocity 1400 723.3 719.8 403.8 58.6 40.4 26.8 21.9 14.1 1600 830.1 833.5 470.6 68.9 43.1 30.8 24.8 15.9 1800 939.6 921.2 526.9 78.4 49.1 34.4 27 18.4 3000 1549.4 1547.7 894.1 131.2 85.5 55.7 47.7 29 3200 1655.7 1660.8 939.9 137.6 86.8 64.1 49.5 31.6 5400 2788.7 2820.3 1600.6 233.2 150 107.2 79.9 49.4 6400 3301.7 3284.4 1862.4 275.7 174.8 121.2 103.2 58.7 9000 4662.9 4645.2 2642.8 380.6 258.2 184.5 128.2 80.2 11000 5684.4 5715.3 3228.5 461.4 308.2 215.7 163.8 105.7 16000 8273.1 8227.5 4689.6 678.6 447.3 312.1 248.7 145.9 18000 9310.7 9355.1 5329.3 754.5 498.4 360.7 277.3 165.3 22400 11652.2 11591 6506.8 971.6 628.3 435.5 351.5 210.3 25200 13044.7 13009.6 7363.8 1091.3 701.7 491.3 387.8 244 40000 20683.6 20728.4 11714.1 1706.5 1108.6 779.4 615.7 378.4 45

Table A.8: Average standard deviation of number of encounters for given areas and velocities when θ = 270, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 1.99 2.5788 1.7436 1.2 0.922 1.1874 0.4583 0.6403 50 4.4057 3.9256 3.527 0.8 1.2806 0.6633 0.6403 0.4 75 2.7203 5.2574 4.4677 1.9079 1.3565 1.4 0.9434 0.6708 100 6.0671 5.8215 3.4583 1.6401 2.1071 1.4967 1.7 0.6633 200 6.9921 10.4096 5.4553 24.5701 2.2 1.9209 1.2689 1.044 300 11.9516 9.4106 6.3914 3.5623 2.3749 2.2891 2.6096 0.7 400 10.9818 12.9804 14.1435 3.9102 3.4943 2.6851 2.1541 1.3454 500 16.5048 21.844 13.6854 3.2924 3.0676 2.3431 2.0591 1.4967 600 13.0465 12.4756 15.174 2.3431 4.5266 4.7127 2.9479 3.0725 700 12.9784 19.3362 8.0405 5.1923 2.7 3.3181 3.0806 2.7276 800 12.6633 22.3564 13.4298 5.4 4.3543 3.4409 2.6382 3.2619 900 21.8963 22.5033 16.3963 4.8208 3.0806 3.2924 3.2496 1.9494 1000 19.1732 23.0957 10.2767 5.748 4.9689 3.3526 3.3302 2.9394 Velocity 1200 20.5611 21.7081 24.2052 7.6 4.4721 5.1039 2.245 2.8302 1400 24.4256 16.4183 13.7099 5.8344 4.3635 3.8158 2.7731 3.7 1600 30.4974 26.4585 16.0325 6.549 8.8595 4.4677 4.6861 3.9102 1800 33.0339 28.5125 22.0202 12.3305 8.3481 4.0792 4.899 3.2 3000 31.4204 43.9182 21.459 12.2049 7.6583 7.5769 5.001 5.3292 3200 35.4938 30.5837 26.0133 11.655 6.9397 6.1555 7.7104 7.4189 5400 46.2602 50.9844 34.4593 11.7796 12.025 8.9755 5.3376 6.7705 6400 70.0401 52.7943 57.9227 15.3886 16.606 14.5038 9.239 8.3193 9000 77.745 71.7744 50.4714 26.5187 13.0138 12.8627 10.7406 7.3185 11000 61.5113 76.0934 63.8471 9.8306 18.503 9.2418 12.2213 12.5304 16000 71.0527 66.6097 73.3147 20.3725 19.9552 19.776 15.7737 8.5376 18000 110.4654 51.8796 88.0114 23.6442 21.8961 13.5798 10.8079 9.6856 22400 114.4262 113.1998 61.1487 30.4178 15.4146 23.8506 12.0768 19.8547 25200 111.4334 118.7335 58.7449 33.8055 36.8864 21.4432 15.7721 16.5409 40000 164.7369 116.6089 66.4687 40.8516 28.7931 23.9633 11.9587 21.8687 46

Table A.9: Average number of encounters for given areas and velocities when θ = 360, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 12.8 13.2 8.7 1.3 0.4 0.5 0.3 0.1 50 26.3 24.8 12.9 1.8 1.4 0.9 0.8 0.3 75 38 39 21 3.7 1.9 0.9 1.2 0.5 100 58.7 60 27.5 4.2 2.8 1.4 1.6 0.8 200 119.8 118.5 56.7 9 6.1 4 2.9 1.6 300 176.3 176.2 84.9 12.7 9.4 4.1 5.9 3.2 400 205.6 194.2 119.6 15.8 11.6 8.2 7.2 5.8 500 257.7 253.5 144.9 21.4 14.1 10.4 6.6 5.8 600 306.1 310.5 173.3 25.1 17.3 11.9 10.1 6 700 367.8 366.5 200.8 29.3 20.5 12 9.5 6.4 800 407.8 410.2 239.5 31.2 22.2 15.4 11.9 6.9 900 460 462.3 264.7 41.3 25.2 17.2 14.5 7.2 1000 518.1 519.8 293.6 44.1 27.8 19.6 17.3 7.9 Velocity 1200 611.6 626.2 366 50.7 35.4 21.6 18.8 11 1400 728.5 716.1 411.3 62.1 38.5 27.3 21.8 12 1600 818.5 816.3 464.6 68.4 42.4 31.4 26.3 15.8 1800 1039.7 1051.7 586.5 82.8 55.1 38 29.6 18.1 3000 1742.1 1743.4 982.1 142.4 90.3 69.4 54.6 32.3 3200 1879.1 1883.3 1058.9 158.7 105.5 71.4 55.8 31 5400 3153.9 3166.5 1767.6 266.3 167.1 117.5 96.8 58.5 6400 3736 3739.7 2125.5 307.7 199.8 142.8 101.9 68.8 9000 5242.8 5243.7 2996.1 427.1 292 192 158.6 92.2 11000 6431.4444 6452.6667 3625 540 351.6667 245 196.3333 127.1111 16000 9383.9 9376.9 5282.3 775.1 502.5 355.5 276.6 175.7 18000 9306.9 9334.6 5263.1 775.6 497.4 352.7 280.4 168.5 22400 11600.3 11614.5 6594.6 930.8 618.9 441.9 341.3 203.9 25200 13051.4 13093.8 7375.5 1080.7 701.3 500.1 389.8 239.9 40000 20731.5 20736.2 11762.4 1702.9 1117.7 781.8 616.4 385.7 47

Table A.10: Average standard deviation of number of encounters for given areas and velocities when θ = 360, R = 0.05, and T = 1 day.

Area 25 35 50 400 625 900 1200 2000 25 3.6551 2.7495 2.0025 0.9 0.4899 0.6708 0.4583 0.3 50 5.1 4.7707 3.2388 1.4 1.0198 0.9434 0.9798 0.6403 75 7.3348 4.4944 3.7417 1.1 1.4457 1.2207 0.9798 0.6708 100 9.4979 10.2274 5.3898 1.99 2.0396 1.2 0.4899 0.7483 200 9.9579 9.2763 7.0434 2.3238 2.4269 1.6125 1.044 1.3565 300 15.2384 11.435 8.0181 2.9343 2.9052 1.6401 3.2078 1.8868 400 12.1012 7.427 13.3357 3.2496 2.245 2.7857 2.358 2.358 500 10.479 9.9725 8.2517 4.128 4.5266 2.245 2.5377 1.9391 600 15.0562 12.4036 11.6623 4.0853 2.8302 2.508 3.6455 2.4083 700 14.3722 13.8726 20.2129 3.743 4.3875 3.3764 3.3242 3.1686 800 17.1278 17.3482 15.7242 4.5343 4.1905 3.1369 3.0806 3.5057 900 16.4073 20.0102 11.9084 3.9509 4.4 2.358 3.7216 2.0881 1000 29.7908 20.0589 14.5203 7.1896 5.7411 4.128 5.4964 3.0806 Velocity 1200 26.8894 22.6265 19.99 7.4572 4.6519 4.5651 4.8539 2.8983 1400 14.5757 21.375 15.1 7.4626 7.0036 2.9343 5.8275 3.0659 1600 16.5182 22.6232 11.1463 7.3783 7.7356 3.6932 4.981 4.996 1800 22.8825 26.2376 18.8109 5.8275 9.2136 5.3104 7.3376 2.2113 3000 58.3189 32.259 30.5563 10.707 8.6377 5.4809 7.3103 4.4956 3200 32.0701 34.2638 38.7439 6.8272 10.2981 9.1455 6.9397 4 5400 39.4042 52.4161 47.2381 13.2367 12.2919 13.208 6.9685 6.249 6400 66.3355 39.2455 51.7711 12.8222 11.9398 12.9213 14.6181 8.5065 9000 45.5781 62.0807 58.6131 27.844 12.5379 9.716 15.0147 9.8265 11000 70.1461 39.6933 29.5935 23.4426 22.6176 15.0185 12.0738 10.7749 16000 82.2465 102.8712 83.3847 15.8899 18.1122 17.3277 11.8423 15.1066 18000 115.2349 110.9137 40.7295 28.4823 15.4674 10.7336 15.7873 12.8627 22400 104.8905 114.2762 102.5214 29.556 22.2056 23.6409 18.2047 13.5827 25200 181.0139 99.016 67.7942 31.6166 31.0227 17.6717 22.8858 10.3581 40000 93.8033 183.013 130.4486 43.1774 28.1249 20.3165 40.8808 13.9 48

Table A.11: K values for each θ calculated from data found in Tables A.1-A.10. Kup and Kdown are calculated from equation 2.3 using encounter rates ± 1 standard deviation. The most variation in K occurs when θ = 0 (individuals remain on a straight path with no change in direction). Otherwise variation in K is less than 0.01.

θ = 0 θ = 90 θ = 180 θ = 270 θ = 360

Kup 0.1307 0.1235 0.1236 0.1235 0.1262 K 0.1231 0.1225 0.1225 0.1224 0.125

Kdown 0.1155 0.1214 0.1213 0.1213 0.1238 49

Appendix B

NetLogo Code for Pair Formations

extensions [sound] globals [encounters counter]

;;;;;;;;;Declare the different brands of agents ;;;;;;;;; breed [females female] breed [males male] turtles-own [energy] patches-own [nummales numfemales]

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;THESETUPBUTTON;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to setup set counter 0 setup2 end to setup2 clear-ticks clear-turtles clear-patches clear-drawing clear-all-plots clear-output reset-ticks ;; create the world based on insect step size resize-world 0 sqrt(world-size) 0 sqrt(world-size) set-default-shape turtles "butterfly" set encounters 0

;;;;;;;;;Create Females ;;;;;;;;; create-females num-females [ 50

set size .001 set color yellow set energy (24 * velocity * lifespan) setxy random-xcor random-ycor ]

;create normal males create-males num-males [ set size .001 set color blue set energy (24 * velocity * lifespan) setxy random-xcor random-ycor ] end

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;;;THEGOBUTTON;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to go tick ask females [ female-search set energy energy - 1 death ]

ask males [ males-search set energy energy - 1 death ]

if ((count males + count females) = 0) [file-open "T=1 day, R=.5, theta=0 v=18000 A=35.csv" file-write encounters file-print "," file - close set counter counter + 1 ifelse counter < 10 [ setup2 go] [ stop ] ] end 51

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;FEMALESEARCHING;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to female-search rt (random angle) lt(random angle) fd 1 end

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;;;MALESEARCHING;;;;; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; to males-search rt (random angle) lt (random angle) fd 1 let mate one-of females in-radius .5 if mate != nobody [set encounters encounters + 1 hatch 1 [setxy random-xcor random-ycor] die ] end

;;;;;;;;;;;;;;;; ;;;DEATH;;;;; ;;;;;;;;;;;;;;;; to death if energy < 0 [die] end 52

Appendix C

Matlab Code for Pest Control Model

function RatInt K1 =800; K2 =1600; Time=130; %months That1 =500; That2 =500; tr =1/30; population1=255; population2=844; z1=population1/2; z2=population2/2; sol = dde23(@ratintEQ, [3+22*tr 23*tr 3+22*tr 23*tr], @ratinthist, [0,Time]); %lag times for t_in1, t-t_r [3months+23days, 23days, 3months+23days, 23days] figure (1); subplot(2,2,1) plot(sol.x, sol.y(1:5,:)) set(gca,’FontWeight’,’Bold’,’FontSize’,11) title(’Rat fertilization management patch \Pi_1’); xlabel(’time (months)’); ylabel(’Rat population density’); subplot(2,2,3) plot(sol.x, sol.y(6:10,:)) set(gca,’FontWeight’,’Bold’,’FontSize’,11) title(’Rat fertilization management patch \Pi_2’); xlabel(’time (months)’); ylabel(’Rat population density’); subplot(2,2,2) plot(sol.x,sol.y(1,:)+sol.y(2,:)+sol.y(3,:)+sol.y(4,:)+sol.y(5,:)) set(gca,’FontWeight’,’Bold’,’FontSize’,11) title(’Patch \Pi_1’); xlabel(’time (months)’); ylabel(’Rat population density (total)’); 53

subplot(2,2,4) plot(sol.x,sol.y(6,:)+sol.y(7,:)+sol.y(8,:)+sol.y(9,:)+sol.y(10,:)) set(gca,’FontWeight’,’Bold’,’FontSize’,11) title(’Patch \Pi_2’); xlabel(’time (months)’); ylabel(’Rat population density (total)’);

function s=ratinthist(t) %Constant history funtion. s = [z1 (2*z1)/3 0 z1/3 0 z2 (2*z2)/3 0 z2/3 0]’; %starting populations for M1, F1, Fs1, Fr1, Ms1, M2, F2, Fs2, Fr2, Ms2 end function dydt = ratintEQ(t,y,Z) %Differential equations function. mu =1/5; delta =.5; lambda =8; alpha =0.3; E =0.5; rho1 = 2; rho2 = 2; gamma1 = 2; gamma2 = 2; ylag1 = Z(:,1); ylag2 = Z(:,2); ylag3=Z(:,3); ylag4=Z(:,4); M1 = max(0,y(1)); F1 = max(0,y(2)); Fs1 = max(0,y(3)); Fr1 = max(0,y(4)); Ms1 = max(0,y(5)); M2 = max(0,y(6)); F2 = max(0,y(7)); Fs2 = max(0,y(8)); Fr2 = max(0,y(9)); Ms2 = max(0,y(10));

T1=max(0,M1+F1+Fs1+Fr1+Ms1); T2=max(0,M2+F2+Fs2+Fr2+Ms2); dF12 = rho1*F1*(((T2-That2)^2)/(1+((T2-That2)^2))); dF21 = rho2*F2*(((T1-That1)^2)/(1+((T1-That1)^2))); dFs12 = rho1*Fs1*(((T2-That2)^2)/(1+((T2-That2)^2))); dFs21 = rho2*Fs2*(((T1-That1)^2)/(1+((T1-That1)^2))); dM12 = gamma1*M1*(((T2-That2)^2)/(1+((T2-That2)^2))); 54 dM21 = gamma2*M2*(((T1-That1)^2)/(1+((T1-That1)^2))); dMs12 = gamma1*Ms1*(((T2-That2)^2)/(1+((T2-That2)^2))); dMs21 = gamma2*Ms2*(((T1-That1)^2)/(1+((T1-That1)^2))); chi_in1 =(ylag1(1)*ylag1(2) + ylag1(5)*ylag1(2) + + ylag1(1)*ylag1(3))/((ylag1(1)+ylag1(5))*(ylag1(2)+ylag1(3))); phi_in1 = E * min(ylag1(1) + ylag1(5), ylag1(2) + ylag1(3)); chi_in2 =(ylag3(6)*ylag3(7) + ylag3(10)*ylag3(7) + + ylag3(6)*ylag3(8))/((ylag3(6)+ylag3(10))*(ylag3(7)+ylag3(8))); phi_in2 = E * min(ylag3(6) + ylag3(10), ylag3(7) + ylag3(8)); chi_t1=(M1*F1 + Ms1*F1 + M1*Fs1)/((M1+Ms1)*(F1+Fs1)); phi_t1= E * min(M1 + Ms1, F1 + Fs1); chi_t2=(M2*F2 + Ms2*F2 + M2*Fs2)/((M2+Ms2)*(F2+Fs2)); phi_t2= E * min(M2 + Ms2, F2 + Fs2); dM1dt = (-mu*M1+delta*lambda/2*(chi_in1*phi_in1)+dM21-dM12)* * (1-T1/K1)-(1-exp(-alpha*M1))*M1; %M1 dF1dt = (-mu*F1+delta*lambda/2*chi_in1*phi_in1+dF21-dF12)* * (1-T1/K1)-chi_t1*phi_t1+ylag2(4)-(1-exp(-alpha*F1))*F1; %F1 dFs1dt = (1-exp(-alpha*F1))*F1+(-mu*Fs1+dFs21-dFs12)*(1-T1/K1); %Fs1 dFr1dt = chi_t1*phi_t1-ylag2(4)-mu*Fr1*(1-T1/K1); %Fr1 dMs1dt = (1-exp(-alpha*M1))*M1+(-mu*Ms1+dMs21-dMs12)*(1-T1/K1); %Ms1 dM2dt = (-mu*M2+delta*lambda/2*(chi_in2*phi_in2)+dM12-dM21)*(1-T2/K2); %M2 dF2dt = (-mu*F2+delta*lambda/2*(chi_in2*phi_in2)+dF12-dF21)*(1-T2/K2) - - chi_t2*phi_t2+ylag4(9); %F2 dFs2dt = (-mu*Fs2+dFs12-dFs21)*(1-T2/K2); %Fs2 dFr2dt = chi_t2*phi_t2-ylag4(9)-mu*Fr2*(1-T2/K2); %Fr2 dMs2dt = (-mu*Ms2+dMs12-dMs21)*(1-T2/K2); %Ms2 dydt = [dM1dt dF1dt dFs1dt dFr1dt dMs1dt dM2dt dF2dt dFs2dt dFr2dt dMs2dt ]; end end 55

Appendix D

Python Code with GIS to Predict Rat Populations

#rat population prediction

#import modules import pandas as pd #import numpy as np import arcpy

#name workspace folder workspace = r’C:\Users\Katherine\Desktop\rats-gis’ arcpy.env.workspace = workspace arcpy.env.overwriteOutput = True #insert data # ****write path and filename of data to use**** data = pd.read_csv(r’C:\Users\Katherine\Desktop\Box Sync\Box Sync\Flashdrive_12-21 \Python\Project\Data\311_Service_Requests_for_2009_NYC.csv’, header=0) #output file names # ***name ouptput files and paths to use**** shp_file = r’C:\Users\Katherine\Desktop\rats-gis\NYC2009d.shp’ shp_file_name = ’NYC2009d.shp’ rast_file_name = r’C:\Users\Katherine\Desktop\rats-gis\NYC2009d.tif’ #choose column for description # ****header labels in data**** desc = ’Descriptor’ #choose column for latitude lat = ’Latitude’ #choose column for longitude lon = ’Longitude’ types = data[desc].unique() #print types #use if need to identify types #NYC 2009 - ’Condition Attracting Rodents’ ’Signs of Rodents’ ’Rat Sighting’ # ’Mouse Sighting’ ’Rodent Bite - PCS Only’

#****choose number to multiply by for each category**** multiplier = [1, 2, 3, 0, 1] 56

#create empty array for multipliers in order of data mult = [] #iterate through rows and add multiplier in order to empty array # ****need to change if size of multiplier changes**** for index, row in data.iterrows(): if row[desc] == types[0]: mult.append(multiplier[0]) elif row[desc]== types[1]: mult.append(multiplier[1]) elif row[desc]==types[2]: mult.append(multiplier[2]) elif row[desc]== types[3]: mult.append(multiplier[3]) elif row[desc]==types[4]: mult.append(multiplier[4])

#add multiplier to new column in data data.loc[:,’Multiplier’] = mult

#delete all rows with Multiplier=0 data=data[data.Multiplier != 0]

outdata = data[[desc, ’Multiplier’, lat, lon]]

#print outdata

#Create Shapefile #define spatial reference to use sr = arcpy.SpatialReference(’NAD 1983’) #Create shapefile with spatial reference arcpy.CreateFeatureclass_management(workspace,shp_file_name,’MULTIPOINT’, spatial_reference=sr)

#add type field arcpy.AddField_management(shp_file_name,’DESCRIPTOR’,’TEXT’) #add population field arcpy.AddField_management(shp_file_name,’POPULATION’,’LONG’)

with arcpy.da.InsertCursor(shp_file_name, [’id’,’DESCRIPTOR’,’POPULATION’, ’shape@’]) as inserter: for row in outdata.values: dsc = row [0] pop = row [1] xcor = float(row[3]) ycor = float(row[2]) 57

#print type(xcor), type(ycor) #create points array pts = arcpy.Array() #add coordinates to pts array pts.add(arcpy.Point(xcor,ycor)) #create point geometry point = arcpy.Multipoint(pts) #list id, description, population, and multipoint x = [1, dsc,pop,point] #insert x into shapefile inserter.insertRow(x)

#create point to raster parameters inFeatures = shp_file_name valField = ’POPULATION’ outRaster = rast_file_name assignmentType = ’SUM’

#execute point to raster arcpy.PointToRaster_conversion(inFeatures, valField, outRaster, assignmentType)