Smith’s Population Model in a Slowly Varying Environment

Rohit Kumar

Supervisor: Associate Professor John Shepherd

RMIT University

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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

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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

The Malthusian model (1) generated solutions ( )N T 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,

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

0, (0) . (1)dN

RN N NdT

01 , (0) . (2) dN N

RN N NdT K

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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,

where

This system is autonomous as all the parameters are independent of time, T . All

is the Malthusian growth constant

is Smith’s constant

is the carrying capacity

is the population density

 

R

C

K

N

0

1

, (0) . (3)

1

N

dN KRN N N

NdTC

K

4

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:

Using exponential properties, we see that the left hand side of (4) tends to zero

so that, for consistency, we must have N K 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: , and .R C K

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.

0

0

1

( ) (4)

1

K

RT C

K

Ne K NK

N

5

In Figure 1, all the curves commence at the initial population, 0N . As seen in the

graph, the curves tending towards the capacity value of 1. Also, as 0N K , the right

hand side terms of the implicit solution become smaller, hence, the exponential decay.

0Case 1: N K

0Case 2: N K

Figure 1: Constant coefficient when 0.50.C

Figure 2: Constant coefficient when 0.75.C

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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

, and R K C 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 , and R K C vary with time, the system is no longer autonomous as

the variables are dependent on time, as below:

Here, the differential equation is not usually exactly solvable and must be solved

numerically, in general. However, in many cases, , and R K C 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

0

0 *

0

0 *

0

0 *

0

0 *

0

(6)

1( ) (7)

1( ) (8)

1( ) = (9)

1( )

t R T

R t R r tR T

N t K n tR T

K t K k tR T

C t C c tR T

(10)

0

( )1

( )( )( ) ( ) , (0) . (5)

( )1 ( )

( )

N T

K TdN TR T N T N N

dT N TC T

K T

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where t is non-dimensional time, n is a variable and , and r c k are dimensionless

parameters that vary with time. The characteristic values of the parameters are given by

0 0 0, R K Cand whilst *T is a characteristic time scale (Roberts, 2014). We define by

that is, the ratio of the time scale of N to the time scales for , and R C K . Hence, (3)

can be written in non-dimensionalised form as

If is small, then the parameters, , and r c k are varying slowly with different

time scales, compared to the population, n . Thus, two time scales will be introduced;

‘normal time’, denoted by 0 1

1( )t g t

and ‘slow time’, denoted by 1t t where 1( )g t

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 0t and 1 :t

The derivative of n is obtained by using the chain rule to give the following

expression:

Here, both 0D and 1D are partial derivatives taken with respect to 0t and 1t respectively.

0

* *

0

11 = (11)

R

R T T

0

0 0

( )1

( ) ( )( ) ( ) , (0) . (12)

( ) ( )1

( )

n t

Ndn t k tr t n t n

c c t n tdt K

k t

0 1( , ) ( , , ) n t n t t

0 1

0 1

0 1 1

1 0 1

1 ( '( ) )

'( ) (13)

t tdn n n

dt t t t t

D n g t D n

g t D n D n

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Now, the initial value problem converts to:

which is the multi-scale problem. Note that the differential equation in (14) is now a

partial differential equation.

The Multi-scaling method is a technique that calculates an approximate solution

for the single species population model by inducing fast time and ordinary time for

independent variables (Nayfeh, 1973). It is a technique that has been commonly used in

the Verhulst model and will be used here in Smith’s model. It has also been used for

logistic population models that have carrying capacities varying slowly (Grozdanovski

et. al, 2009). The multi-scale problem can be solved using the perturbation approach.

This approach involves substituting the perturbation expansion below into the

differential equation in (14), after which some algebraic manipulation is performed and

like powers of ‘ ’ are collected.

The perturbation expansion is broken down into problems for the initial

approximation,0n , the higher order approximation,

1n and further terms.

Once, we substitute (15) into (14), and equate powers of , we obtain the

differential equations for both 0n and

1n as shown below:

Solving (16) provides an implicit solution for 0n . The implicit solution is

1 0 10

( )1

( )'( ) ( ) ( ) , (0,0, ) . (14)

( )1

( )

n t

k tg t D n D n r t n t n

c c t

k t

2

0 1 0 0 1 1 0 1( , , ) ( , ) ( , ) ( ) (15)n t t n t t n t t

0

0 0 0 00 0

0 1 1 1 0 1 0

0

1

' (16)

1

g'D , (17)1

n

kg D n rn nc cn

k

rn n D n n n

c c

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where , ', , and D g c r k are arbitrary functions of 1t , 0c is a constant, 0t is normal time

and 0n is the leading order approximation.

The elimination of secular terms in (17) leads to the expression for 0t as given

below:

As 0t , it is found from (18) that

Using initial conditions, 0 1 00, t t n to solve for the constants and then

eliminating secular terms, which are terms that play a role in the approximation

growing unboundedly, the full implicit solution for 0n is

It is this implicit solution that was used in plotting the multi-timing

approximations below for parameters that vary slowly. One of the parameters that we

chose to look at varying slowly is (Smith's constant)c . This will be discussed in the

next section.

1 00

1

( )1'( )0 1

1

0

( ( ))( ) (18)

r t tc cg tn k t

D t en

000

11

1 (0)10

0

(0)( (0)) (20)

c cc ctn k

n k k ek

1 0

1 0 10 1

( )1

'( )(1 ( ))1 ( )

0 1 1 1( ) ( ( ) ( )) .

r t t

g t c c tc c tn k t k t D t e

00

0

1 (19)

1 ( )

t rt ds

c c s

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Figure 3: Graph for varying 1 sin( ) when 0.2,

0.1, 0.25.

c t µ

and

Figure 4: Graph for varying when 1, 1.5,

1

0.85, 0.2 and 0

.001.µ

ac a b

tbe

7. Smith’s parameter, c , varying slowly with time using the Multi-

timing approximations

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Figure 5: Multi-timing approximations versus numerical solution graph

when 0.2, 1, 0, 10

0.05, 0.2, and 1 ( ( )).

µ c c r

k sin t

The next case is when c varies slowly with time. c varying slowly with time is of

interest as it is responsible for modifying the overall growth rate of Smith’s model.

In Figure 3, we observe exponential growth. The graph illustrates that all three

curves reach the carrying capacity value of 1k at slightly different rates subject to the

initial c values chosen. As the initial c values become larger, the solution reaches its

limit a bit slower. All curves commenced at 0.2 as (0) 0.2n was the initial condition

defined.

In Figure 4, exponential growth is also observed with a differential initial

condition value of 0.85 . Similar to Figure 3, there is some discrepancy in all the

curves as the initial c values becoming larger. This may be due to similar variation in

the coefficients. As ,1 t

ac

be

the curves approach 1a .

8. Comparison of Multi-Timing Approximations with Numerical

Solutions

Part of examining the performance of Smith’s Model required comparing the multi-

timing approximations when the parameters , and R C K vary slowly, with numerical

solutions that were generated using Maple. This is equivalent to testing this for the

dimensionless parameters r, c, and k. The Figures below show both the multi-timing

approximations and numerical solutions.

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Figure 6: Multi-timing versus numerical solution graph

when 0.5, 1, 0, 1 , 0

0.0

( (

1, 0.0

))

5 1. 

µ c c r sin t

and k

Figure 7: Multi-timing versus numerical solution graph when

0.7, 1, 0, 0.01, 0.1, 0

1, 1 and 1( )  ( ) .

µ c c

k sin t r

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Figure 8: Multi-timing versus numerical solution graph when

0.4, 0.4, 1, 0, 0

0.15, 0.3, 1 and( ) 1.

µ a c c

tk ae r

Figure 5 illustrates oscillations as 1 sin( )k t for reasonably small and values.

The graphs commence at 0.2 as the initial condition was set to (0)n . It always

helped to make the values less than 0.2 in order for the graphs to be plotted over a

large t -range. The multi-timing approximation and the numerical solution graphs match

up very well with each other and is the case when r and k are set to exponential

growth/decaying functions.

However, Figure 6 shows lack of oscillations as 1 sin( )r t for again

reasonably small epsilon and delta values. Lack of oscillations may be due to the fact

that r appears in (20), and nowhere else. Therefore, its effect is an integrated one, and

oscillations are summed out, so don't appear in 0n . On the other hand, k appears outside

0t , so oscillations appear in 0n .

Similarly, in Figure 5, we see oscillations occurring in Figure 7 as

1 sin( ).k t The only difference is there is an placed inside the sine function.

However, the oscillations are taking place a bit quicker perhaps due to a larger value

chosen as opposed to that in Figure 5.

Figure 8 on the other hand, illustrates an exponential growth as expected. The

exponential function that k is set to tends to 1 as t approaches infinity.

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All four graphs illustrate that there is minor discrepancy between the multi-

timing approximations and numerical solutions, suggesting that the approximation fits

well with the numerical solution.

9. Conclusion

Overall, the Smith’s model has proven to be an accurate fit for various cases that were

mentioned earlier especially for the numerical solution versus multi-timing

approximation scenario. The constant coefficient Smith’s model has been extended to

one involving slowly varying parameters, using the multi-scaling approach. We observe

slower change in the coefficients as the epsilon values become smaller. It helps to

extend the perturbation expression to higher powers of for large values of .

References

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logistic model with slowly varying coefficients’, Applied Mathematics Letters,

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