Handout on Theoretical Models for Predator-Prey Population Dynamics

Dirk Metzler October 9, 2018

Contents

1 A simple predator-prey model by Volterra 1

2 Should we believe in these oscillations? 6 2.1 Intraspeciﬁc competition ...... 6 2.2 When predators eat only as much as they can ...... 8 2.3 Conclusions ...... 10

References

[BTH05] M. Begon, C.R. Townsend, J.L. Harper (2005) Ecology — From Individuals to Ecosystems, 4th Ed.Blackwell Publishing

[K01] M. Kot (2001) Elements of Mathematical Ecology Cambridge University Press [C00] T.J. Case (2000) An Illustrated Guide to Theoretical Ecology Oxford University Press

I will refer to several Figures in Chapter 10 [BTH05]. I do not include these Figures in this handout for several reasons: 1. to keep the handout small in terms of megabytes and printed pages 2. to avoid copyright issues

3. as it is a good idea to read this and other chapters in the text book

4. as the ﬁgures can be downloaded from http://www.blackwellpublishing.com/begon Before we look at theoretical models, remeber:

• Essentially, all models are wrong, but some are useful. (George E.P. Box) • Begon et al. (2005, p. 297): “In fact, simple models are most useful when their predictions are not supported by real data – as long as the reason for the discrepancy can subsequently be discovered.”

1 A simple predator-prey model by Volterra

Frequency of selachia (old name for sharks and simlar, mostly predators) on several italian ﬁsh markets observed by marine biologist Umberto D’Ancona:

1 40 ●

30 ●

● ● ● 20 ● ● ● ● ● 10 0 Percentage of selachians Percentage

1914 1916 1918 1920 1922

Year

• Almost no ﬁshing from 1914 to 1918 (World War I) • Why does reduced ﬁshing lead to an increase of predators relative to prey?

D’Ancona’s father in law, the mathematician Vito Volterra tried to answer this question with a simple mathematical model: N(t) prey population size

P (t) predator population size

dN = rN − cP N = N · (r − cP ) dt dP = bNP − mP = P · (bN − m) dt • If P ≡ 0, N grows exponentially

• If N ≡ 0, P shrinks exponentially • eﬀects of N and P to each other are mass-action terms For the analysis, get rid of two parameters by changing variables: b x := N m c y := P r ⇒ dx = r(1 − y)x dt dy = m(x − 1)y dt

2 r= 1 , m= 1 dx = r(1 − y)x 3.0 dt

dy 2.5 = m(x − 1)y dt 2.0

3.0 x y

y 1.5 2.5 2.0 1.0 1.5 1.0 Scaled population sizes 0.5 0.5 0.0

0 10 20 30 40 0.0

Time 0.0 0.5 1.0 1.5 2.0 2.5 3.0

r= 1 , m= 1 r= 2 , m= 2 3.0 3.0 2.5 2.5 2.0 2.0 y y 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

x x

r= 1 , m= 2 r= 2 , m= 1 3.0 3.0 2.5 2.5 2.0 2.0 y y 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

x x

dx dy = r(1 − y)x = m(x − 1)y dt dt

3 r=2

3.0 x y 2.5 2.0 1.5 1.0 Scaled population sizes 0.5 0.0

0 10 20 30 40

Time

r= 1 , m= 1 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Helpful for the anaylsis of the of the system are the

x ﬁxed points: points y with dx dy = = 0, dt dt that is, if the system starts there, it will stay there isoclines: lines (or curves) on which either

dx dy = 0 or = 0 dt dt or both

ﬁnding isoclines and ﬁxed points dx 0 = = r(1 − y)x ⇒ x = 0 or y = 1 dt

4 dy 0 = = m(x − 1)y ⇒ x = 1 or y = 0 dt

r= 1 , m= 1 3.0

2.5 Fixed points: x 0 2.0 = y 0 y 1.5 and 1.0 x 1 = 0.5 y 1 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Stability of these two fixed points Remeber that local stability means that if the system starts very close to the fixed point it will move to it. 0 Is 0 locally stable? dx dy ≈ rx ≈ −my dt 0 dt 0 ( 0 ) ( 0 ) It is not stable as it will be left into direction of x (leading to exponential growth of host population). 1 The mathematical analysis of 1 is a bit more subtle, see [K01]. In fact, if the system starts close to this fixed point, it will stay on an orbit around it.

r= 1 , m= 1 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Note that on each orbit, movement is faster for higher values of x and y. Volterra could show that on each orbit, when averaged over time:

average(x) = 1 and average(y) = 1

Thus, for the average abundance of predator or prey over a longer time period, the starting point is

5 irrelevant. Average frequency in original scaling (ﬁxed point): dN = N · (r − cP ) dt dP = P · (bN − m) dt

m r average(N) = , average(P ) = b c Model eﬀect of ﬁshing by additional death rate: dN = N · (r − cP − d) dt dP = P · (bN − m − d) dt

m + d r − d average(N) = , average(P ) = b c

Thus, Volterra’s simple model even predicts, that ﬁshing increases the abundance of prey ﬁsh!

Maybe, this model can explain why the usage of pesticides sometimes lead to an increase in the abundance of the pest.

2 Should we believe in these oscillations?

Three questions: 1. Do we observe long-term oscillations in nature? See Figure 10.1 in [BTH05] 2. Does the model fit these data? E.g. in the hare-lynx system the hare is also prey of other predators and predator of its plant food. This makes thorough statistical ananlysis very complex. Time- series analysis of Lynx abundance given hare data fits well to predator-prey model. GENERAL PROBLEM: Data can never confirm that a model is valid! All models are wrong! 3. Are these osciallations still predicted if we make the model a bit more realistic?

2.1 Intraspeciﬁc competition Lotka-Volterra model for predator-prey with intraspeciﬁc competition: dN = N · (r − cP − kN) dt dP = P · (bN − m − gP ) dt non-trivial isoclines are given by r − kN P = c bN − m P = g

If non-trivial equilibrium with P 6= 0 exists, it must be gr + cm br − mk (N,P ) = , cb + gk cb + gk

6 Thus, coexistence is only possible if br > mk or, equivalently, b k > m r

What if P = 0 and N 6= 0? In this case dN = N · (r − cP − kN) = N · (r − kN) dt and we obtain the ﬁxed point N = r/k. dN r Furthermore: If 0 < N < r/k then dt = N · (r − kN) > N · (r − k k )) = 0. dN r If N > r/k then dt = N · (r − kN) < N · (r − k k )) = 0. r Therefore, (N,P ) = (r/k, 0) is a stable ﬁxed point and k is the carrying capacity of the prey population if there are no predators, that is the population size to which the system will converge, independently of the initial size N > 0.

3.0 N P 2.5 2.0 1.5 Population sizes Population 1.0 0.5 0.0

0 20 40 60 80 100

Time

r= 2 , c= 2 , m= 1 , k= 0.1 , b= 1 , g= 0.1 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

7 r= 1 , c= 1 , m= 1 , k= 1 , b= 1 , g= 1 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.2 When predators eat only as much as they can Model for type II functional response:

dN N cNP = rN 1 − − dT K a + N dP bNP = − mP dT a + N rescaling the variables and setting a 1 m b K N = ax P = r y T = t α = β = γ = c r b r a leads to dx x y = x · 1 − − dt γ 1 + x dy x = β · − α · y dt 1 + x

8 type II model, a=0.7, b=1.5, g=3 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

type II model, a=0.3, b=1.5, g=3 3.0 2.5 2.0 y 1.5 1.0 0.5 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Watch the simulation in the R script! dx isocline dt = 0: x x = 0 y = (1 + x) 1 + γ isocline dy = 0: dt α y = 0 x = 1 − α

9 non-trivial ﬁxed point is stable if and only if

α x > arg max(1 + x) 1 + 1 − α x γ

See Figures 10.6 to 10.12 in [BTH05] Other reasons why oscillations can be damped: • spatial structure: local oscillation are averaged

• role of stochasticity?

2.3 Conclusions Our aims:

• understand ecological mechanisms theoretically/mathematically • but also intuitively/biologically • how intuitive and theoretical predictions correspond to each other

• how robust these predictions are against violations of basic assumptions

Some of the thing you should be able to explain:

• Meaning of diﬀerential equations and how to do calculations • Meaning of each parameter in Lotka-Volterra equations and their generalizations

• How to add a certain mechanisms to Lotka-Volterra equations • How to calculate isoclines and ﬁxed points • Local stability of ﬁxed points and how to analyse it

• What can damp oscillations in predator-prey dynamics • Consequences of diﬀerent functional response assumptions