Smith's Population Model in a Slowly Varying Environment

Smith’s Population Model in a Slowly Varying Environment Rohit Kumar Supervisor: Associate Professor John Shepherd RMIT University CONTENTS 1. Introduction 2. Smith’s Model 3. Why was Smith’s Model proposed? 4. The Constant Coefficient Case 5. The Multi-Scaling Approach 6. The Multi-Scale Equation and its implicit solution using the Perturbation Approach 7. Smith’s parameter, c , varying slowly with time using Multi- Timing Approximations 8. Comparison of Multi-Timing Approximations with Numerical Solutions 9. Conclusion 1 1. Introduction Examples of the single species population models include the Malthusian model, Verhulst model and the Smith’s model, which will be the main one used in our project (Banks 1994). The Malthusian growth model named after Thomas Malthus illustrates the human population growing exponentially and deals with one positive parameter R , which is the intrinsic growth rate and one variable N (Banks 1994). The Malthusian growth model is defined by the initial-value problem dN RN, N(0) N . (1) dT 0 The Malthusian model (1) generated solutions NT()that were unbounded, which was very unrealistic and did not take into account populations that are limited in growth and hence, the Verhulst model was proposed (Bacaer 2011; Banks 1994). This model suggested that while the human population nurtured and doubled after some time, there comes a stage where it tends to steady state (Bacaer 2011). The Verhulst model is defined by the initial-value problem, dN N RN1 , N(0) N0 . (2) dT K Here, R is the intrinsic growth rate, N is the population size and K is the carrying capacity. In other words, the carrying capacity is the biggest population size an environment withstands for an indefinite period. The Verhulst model was also used to model the population of reindeer in Alaska, which grew exponentially until 1938 after which it tended to a limit (Pianka 2000). One of the problems to which the Verhulst model was applied was that of yeast growth. It was noted that this problem was too complicated to be compared with equations that have random constants (Smith 1952). The Verhulst model measures population of a country over a period of time. For instance, it measured the population of Belgium from 1700 to 2000 and found it to be exponentially increasing (Bacaer 2011). The model was useful for extrapolating the population of Belgium from 1851 onwards and also the fact that the population values exceed the carrying capacity obtained. The population in the United States was measured by (Pearl and Reed 1920) through a least square regression fit and a logarithmic parabola. The logarithmic and exponential models developed in their paper provide accurate results of past population 2 however, catalysts that are bound to have an impact on increasing the population will be reduced, which is when the Verhulst model is useful (Pearl and Reed 1920). A number of models such as the Gompertz model, the logistic population model with slowly varying capacity, and the Verhulst model have examined the population growth and decline whilst some of them have demonstrated accurate multi-timing solutions for the population species. On the other hand, there are some models such as the Verhulst model not agreeing with the experimental data of water fleas (Smith 1963). Since that no model is a perfect fit and is rather hypothetical as highlighted in Smith (1952), an assessment of Smith’s model will add important value to the logistic population model area. Thus, the goal of my research is to identify the strengths and weaknesses of the Smith’s model by examining three cases: 1.) The role that Smith’s Parameter, C , plays in modifying the behaviour of Smith’s model. 2.) Generating multi-timing approximations of Smith’s model for the evolving population and then using them to help observe effects on the changing population when parameters such as C vary slowly with time. 3.) The behaviour of the multi-timing solutions from 2) as compared with numerical solutions that were generated using Maple. 2. Smith’s model Smith’s model was developed by Frederick E. Smith in 1963 and is an extension of the Verhulst-logistic equation based on the fact it deals with 3 parameters as opposed to 2 for the Verhulst-logistic model (Smith 1963). It is given by the initial value problem, N 1 dN K RN, N(0) N0 . (3) dT N 1 C K where R is the Malthusian growth constant C is Smith’s constant K is the carrying capacity N is the population density This system is autonomous as all the parameters are independent of time, T . All 3 the parameters; R ,C and K and variable N , are non-negative. Unlike the Verhulst-logistic model that can be solved explicitly, Smith’s model must be solved implicitly, that is, T is found as a function of N . Smith’s Model equation is a first order nonlinear differential equation. It was solved using separation of variables, followed by partial fractions and then integration to obtain the implicit solution of the initial value problem in the form: K K 1 RT C N e( K N0 ) (4) K 1 N0 Using exponential properties, we see that the left hand side of (4) tends to zero so that, for consistency, we must have NK on the right hand side asT . 3. Why was Smith’s model proposed? Even though the Verhulst model deals with populations that tend to a finite limit, it does not take into account any time lags and that is why the Smith’s model was developed (Smith 1963): to minimise the distortion in the resulting population density that arises from time lags present. Unlike the Gompertz model and the Verhulst-Pearl logistic model that involves just two parameters, Smith’s model involves three parameters: RCK, and . 4. The Constant Coefficient Case The curves below represent the evolving population in the constant coefficient case, with a focus on the way C modifies the behaviour of Smith’s model. The first graph looks at a case when the initial population density exceeds the carrying capacity whilst the second graph looks at when the initial population density is less than the carrying capacity. Both graphs were plotted implicitly using (4) for various initial population densities and carrying capacity values. 4 Case 1: NK0 Figure 1: Constant coefficient when C 0.50. Case 2: NK 0 Figure 2: Constant coefficient when C 0.75. In Figure 1, all the curves commence at the initial population, N0 . As seen in the graph, the curves tending towards the capacity value of 1. Also, as NK0 , the right hand side terms of the implicit solution become smaller, hence, the exponential decay. 5 On the other hand, when the initial population size is less than K , an exponential growth is observed as shown in Figure 2. Since the carrying capacity limit is 1, all the graphs tend to 1. However, as the initial population size gets larger, the curves tend to reach the limit more slowly. Whilst the constant coefficient case was looked at, in real life, the constants RKC, and in fact vary in an environment with time and this is where the Multi- Scaling approach becomes useful. 5. The Multi-Scaling Approach When the constants vary with time, the system is no longer autonomous as the variables are dependent on time, as below: NT() 1 dN() T KT() RTNTNN( ) ( ) , (0)0 . (5) dT NT() 1 CT ( ) KT() Here, the differential equation is not usually exactly solvable and must be solved numerically, in general. However, in many cases, RKC, and vary slowly with time so multi-scaling methods may be used. 6. The Multi-scale Equation and its implicit solution using the Perturbation Approach To commence the setup of multi-timing approximations, we want to write (3) in the form of non-dimensional parameters. The non-dimensional time, variables and parameters are given by t R0 T (6) 1 R( t ) R0 r* t (7) RT0 1 N( t ) K0 n t (8) RT* 0 1 K( t ) = K0 k* t (9) RT0 1 C( t ) C0 c* t (10) RT0 6 where t is non-dimensional time, n is a variable and r, c and k are dimensionless parameters that vary with time. The characteristic values of the parameters are given by * RKC0, 0 and 0 whilst T is a characteristic time scale (Roberts, 2014). We define by 1 1 R =0 (11) RTT** 0 that is, the ratio of the time scale of N to the time scales for RCK, and . Hence, (3) can be written in non-dimensionalised form as nt() 1 dn() t kt() N r( t ) n ( t ) , n(0) 0 . (12) dtc0 c()() t n t K 1 0 kt() If is small, then the parameters, r, c and k are varying slowly with different time scales, compared to the population, n . Thus, two time scales will be introduced; 1 ‘normal time’, denoted by t g() t and ‘slow time’, denoted by tt where gt() 01 1 1 is a function that is unknown at this point. However, larger values may be a problem to the multi-timing approach. This may be due to larger deviation away from the solution that arises from the discrepancy in the initial value problem. As two time scales have been introduced, the population density function can now be written as a function of both t0 and t1 : n( t , ) n ( t01 , t , ) The derivative of n is obtained by using the chain rule to give the following expression: dn nt n t 0 1 dt t t t t 01 1 D n( g '( t ) ) D n 0 1 1 g'( t1 ) D 0 n D 1 n (13) Here, both D0 and D1 are partial derivatives taken with respect to t0 and t1 respectively.

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