Model Formulation

Beginning with the ``The Idea'' section, let  denote the population at time . Then, for the time interval  the change in population is

>	dP := P(t+dt) - P(t);

>

The exponential growth model assumes the number of births and deaths during  are proportional to the current size of the population and the length of the interval:

>	birth := beta*P(t)*dt; death := delta*P(t)*dt;

The balance law for this situation is

>	q1 := dP = birth-death;

Eqauation (1) from the handout is obtained when we use  and the equation is divided by  :

>	q2 := simplify( eval( q1, beta=k+delta ) ); eq1 := q2/dt;

The differential equation is the limit of this equation as dt approaches 0

>	q3 := limit( eq1, dt=0 );

Note that the left-hand side of this equation stands for the deriviative of P at t. We can convert this to our standard notation as follows:

>	odeE := convert( q3, diff );

>

Step-by-step derivation of the general solution

This differential equation can be solved by separating variables. Here, equation (2) is obtained by dividing the differential equation by :

>	eq2 := odeE / P(t);

>

To reproduce the steps in the solution of this differential equation it is necessary to rewrite P(t) = P. Then, the left-hand side is integrated with respect to P while the right-hand side is integrated with respect to t:

>	q4 := Int( 1/P, P ) = Int( k, t ) + C;

The integrals are evaluated using

>	q5 := value( q4 );

Applying the exponential function to both sides of this equation yields

>	q6 := exp( lhs(q5) ) = exp( rhs(q5) );

Initial conditions and the constant  

If an initial condition, , is given, this information can be used to find a value for . In general,  so that

>	q7 := C = ln( p[0] );

When this information is inserted into the formula for the general solution (and simplified), we find

>	simplify( eval( q6, q7 ) );

>

The dsolve  command

The steps used to find these solutions are combined into a single, very powerful, Maple command: dsolve . This command is used, in general, to attempt to find solutions to a differential equation. For example, the general solution to the exponential growth model can be found with

>	dsolve( odeE, P(t) );

Note that Maple has written the solution in terms of an indpendent parameter, _C1 . This parameter is related to our constant of integration   by _C1  = .

The dsolve  command can also handle initial conditions. For example,

>	ic := P(0) = p[0];

>	dsolve( { odeE, ic }, P(t) );

This is obviously the same result as we obtained by more explicit steps.

Description of the Logistic Model

The fundamental difference between the exponential and logistic models can be explained in terms of the per capita growth equation. For the exponential model the per capita growth rate is constant.

>	per_capE := eq2;

However, in reality, the per capita growth rate decreases as the population increases

     where     and  

This decrease continues until the population reaches its ultimate capacity. That is, assuming the threshold population is ,

  .

The simplest continuous function satisfying these conditions is the linear function passing through the points ( 0,  ) and ( , 0 ).

That is:

>	g := P -> k - k/L * P;

The logistic growth model can be written in per capita form as

>	per_capL := diff(P(t),t)/P(t) = g(P(t));

or as a differential equation in which the (absolute) growth rate is a quadratic function of the current population size

>	odeL := per_capL * P(t);

>

Solution of the Logistic Model

Analysis of the Solution

While the logistic growth model is not much different from the exponential growth model, the logistic solution is significantly more complicated. It is no longer possible to simply look at the solution and know almost everything about the solution. To begin to understand the solution to the logistic model, define a Maple function that returns the solution for a specific value of , , and :

>	solP := unapply( rhs(r9), (k,L,p[0]) );

For example, the solution to the logistic model with intrinsic growth rate , threshold population , and initial population  is

>	solP(1,10,3);

>

One way to use this solution function is to plot a collection of curves with different initial populations.

>	p0_vals := [0, 50, 100, 150, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400];

>	plot( [ seq( solP(0.0005,1000,p0), p0=p0_vals) ], t=0..20000 );

>

Notice that  solutions with the smaller initial values have one inflection point (where the concavity changes from concave up to concave down).  One interesting property of the logistic growth model is that the time  at which the population passes through this inflection point depends on , , and , but the size  of the population at the inflection point depends only on . To better undestand this property we need to work with the second derivative of the solution: P''( ).