ORDINARY DIFFERENTIAL EQUATIONS POWERTOOLLesson 3 -- Application: Exponential and Logistic GrowthProf. Douglas B. MeadeIndustrial Mathematics InstituteDepartment of MathematicsUniversity of South CarolinaColumbia, SC 29208 URL: http://www.math.sc.edu/~meade/E-mail: meade@math.sc.edu Copyright \251 2001 by Douglas B. MeadeAll rights reserved-------------------------------------------------------------------

3.A Exponential Growth and Decay 3.A-1 General Solution 3.A-2 Doubling Time 3.A-3 Half-Life3.B Logistic Equation 3.B-1 General Solution 3.B-2 Carrying Capacity

restart;with( DEtools ):with( plots ):

The general model for exponential growth and decay with rate constant NiMlImtH, isode := diff( x(t), t ) = k * x(t);and initial conditionic := x(0) = A;If NiMlImtH is positive, the equation models exponential growth, whereas if NiMlImtH is negative, it models exponential decay.This equation is separable, and Maple agrees with this classification, as seen withodeadvisor( ode, [separable] );To find the general solution from first principles, divide through by NiMtJSJ4RzYjJSJ0Rw== to separate variables, and then integrate both sides of the resulting equation to obtainimpl_soln := subs( _t=t, map( int, ode/x(t), t=0.._t,continuous ) );Substitution of the initial condition leads toimpl_part_soln := subs( ic, impl_soln );and solving for NiMtJSJ4RzYjJSJ0Rw== givesexpl_part_soln := simplify(op(solve( impl_part_soln, {x(t)} )));Of course, the same result could be obtained frominfolevel[dsolve] := 3:soln2 := dsolve( {ode, ic}, x(t), [separable] );infolevel[dsolve] := 0:and the final simplificationsimplify( soln2 );

The doubling time for a process growing exponentially is the time NiMmJSJ0RzYjJSJkRw== needed for the quantity to double from its original size. Thus, the doubling time is found from the relationship NiMvLSUieEc2IyYlInRHNiMlImRHKiYiIiMiIiItRiU2IyIiIUYt.Implementing that for the solution found above leads to the equationdouble_eqn := subs( t=t[d], x(t[d])=2*x(0), ic, expl_part_soln );whose solution isdouble_time := solve( double_eqn, {t[d]} );One of the important characteristics of the doubling time is that not only is it the time needed for the initial size to double, it is the time needed for the size to double at any point in an exponential process. The ratio of the solution at time NiMsJiUiVEciIiImJSJ0RzYjJSJkR0Yl and at time NiMmJSJ0RzYjJSJkRw== is given byq1 := x(T+t[d])/x(T) = subs( t=T+t[d], rhs(expl_part_soln) ) / subs( t=T, rhs(expl_part_soln) );If NiMmJSJ0RzYjJSJkRw== is the true doubling time, then this ratio simplifies to 2, as is seen viaq1;`` = simplify(eval( rhs(q1), double_time ));

The half-life for a quantity that is decaying according to an exponential model is the time after which exactly half the original amount remains. The half life NiMmJSJ0RzYjJSJoRw== must obey the equationhalf_eqn := subs( t=t[h], x(t[h])=1/2*x(0), ic, expl_part_soln );Solving for NiMmJSJ0RzYjJSJoRw== leads tohalf_time := solve( half_eqn, {t[h]} );Note that NiMyJSJrRyIiIQ== holds for a decaying process. Thus, the half-life for an exponential model with "growth" rate NiMsJCUia0chIiI= is the same as the doubling time for an exponential model with growth rate NiMlImtH.

The general logistic equation is a modification of the exponential model in which the growth is tempered by the factor (NiMsJiUiS0ciIiItJSJ4RzYjJSJ0RyEiIg==). Therefore, the model consists of the ODElogistic_ode := diff( x(t), t ) = A * x(t) * ( K - x(t) );with, for example, an initial condition of the formlogistic_ic := x(0)=X[0];Initially, when NiMtJSJ4RzYjJSJ0Rw== is small, the factor NiMsJiUiS0ciIiItJSJ4RzYjJSJ0RyEiIg== is essentially constant, so the population obeys an exponential growth law. As NiMtJSJ4RzYjJSJ0Rw== increases and approaches NiMlIktH, the factor NiMsJiUiS0ciIiItJSJ4RzYjJSJ0RyEiIg== tends towards zero, so the rate of change, NiMqJiUjZHhHIiIiJSNkdEchIiI=, approaches zero also. Thus, NiMtJSJ4RzYjJSJ0Rw== approaches a constant, and the growth is said to be self-limiting.This ODE is easily separated by the obvious division, resulting insep_log_ode := logistic_ode / (x(t)*(K-x(t)));Integrating from the initial time, 0, to any other time NiMlInRH, yieldspart_log_soln := subs( _t=t, logistic_ic, map(int,sep_log_ode,t=0.._t, continuous) );and solving for NiMtJSJ4RzYjJSJ0Rw== to obtain the explicit solution, givespart_expl_soln := collect(op( solve( part_log_soln, {x(t)} ) ),exp);A quick check that this is, in fact, a solution to the original ODE showsodetest( part_expl_soln, logistic_ode );and, for the initial condition,eval( part_expl_soln, t=0 );

Before discussing the generic properties of the logistic model, it is instructive to use graphical methods to examine a specific example. For parameter values, chooseparam := { A = 1/2, K=3 };so the logistic equation becomesl_ode := subs(param,logistic_ode);Then, the direction field for this model is seen in Figure 3.1.DEplot( l_ode, x(t), t=0..10, x=0..10, arrows=SMALL, title="Figure 3.1" );An equilibrium solution of the differential equation NiMvLSUiRkc2JCUieEclI3gnRyIiIQ== is any solution NiMtJSJ4RzYjJSJ0Rw== for which NiMvLSUjeCdHNiMlInRHIiIh, identically. Hence, an equilibrium solution is a constant solution. Such constant solutions show up on the direction field as horizontal lines, corresponding to the constant value of NiMtJSJ4RzYjJSJ0Rw==.Figure 3.1 suggests that NiMvJSJ4RyIiJA== might be an equilibrium solution. Closer inspection of Figure 3.1 suggests that NiMvJSJ4RyIiIQ== might also be an equilibrium solution. These potential equilibrium solutions can be found analytically by solving the equation NiMvLSUjeCdHNiMlInRHLSUiZkc2IyUieEc= = 0 for its roots. The roots of the equationEQUILIB_EQN := rhs(l_ode) = 0;are found to besolve(EQUILIB_EQN, x(t));Initial conditions that will produce the equilibrium solutions areequil_ic := [ [x(0)=0], [x(0)=3] ];Use of these initial conditions in the DEplot command yields Figure 3.2.equil_plot := DEplot( l_ode, x(t), t=0..5, equil_ic, x=0..10, arrows=SMALL, linecolor=CYAN, title="Figure 3.2" ):equil_plot;Figure 3.3 shows a sample of solutions with initial conditions between the two equilibria.ic1 := [ [x(0)=i/2] $ i=1..5 ]:soln_plot1 := DEplot( l_ode, x(t), t=0..5, ic1, x=0..10, arrows=SMALL, linecolor=GREEN, title="Figure 3.3" ):soln_plot1;Note that all of these solutions are increasing and appear to approach NiMvJSJ4RyIiJA== as NiMlInRH continues to increase.Figure 3.4 shows a sample of solutions with initial conditions above the positive equilibrium.ic2 := [ [x(0)=2*i] $ i=2..5 ]:soln_plot2 := DEplot( l_ode, x(t), t=0..5, ic2, x=0..10, arrows=SMALL, linecolor=BLUE, stepsize=0.1, title="Figure 3.4" ):soln_plot2;These solutions are all decreasing and also appear to approach the NiMvJSJ4RyIiJA== as NiMlInRH increases.The composite plot is appears in Figure 3.5.display( [equil_plot, soln_plot1, soln_plot2], title="Figure 3.5" );To investigate some of the general properties of solutions to the logistic equation, recall that the equilibrium solutions areequil_sol := solve( rhs(logistic_ode)=0, {x(t)} );A quick inspection of the ODE shows that NiMtJSVkaWZmRzYkLSUieEc2IyUidEdGKQ== > 0 when 0 < NiMtJSJ4RzYjJSJ0Rw== < NiMlIktH and NiMtJSVkaWZmRzYkLSUieEc2IyUidEdGKQ== < 0 when NiMtJSJ4RzYjJSJ0Rw== > NiMlIktH. The long-term behavior of the solutions can be determined from the explicit solution to the IVP by evaluating the following limit.limit_size := Limit( part_expl_soln, t=infinity );The limit exists as a finite number only if both NiMlIkFH and NiMlIktH are nonnegative. With these physically reasonable assumptions, the limit isvalue( limit_size ) assuming A>=0, K>=0;Because all solutions with NiMqKC0lInhHNiMiIiEiIiIlIj5HRiglIktHRig= decrease to NiMvJSJ4RyUiS0c= and all solutions with 0 < NiMtJSJ4RzYjIiIh < NiMlIktH increase to NiMvJSJ4RyUiS0c=, the parameter NiMlIktH is known as the carrying capacity of the logistic equation.The equilibrium solution NiMvJSJ4RyUiS0c= is called a stable equilibrium because, as NiMlInRH increases, nearby solutions, those close and on either side of it, tend towards the equilibrium. This is shown very well in Figure 3.5.The equilibrium solution NiMvJSJ4RyIiIQ== is called an unstable equilibrium because, as NiMlInRH increases, nearby solutions, those close and on either side of it, move away from the equilibrium. Figure 3.6, exhibiting solutions starting both above and below the equilibrium solution NiMvJSJ4RyIiIQ==, shows that this equilibrium is unstable.DEplot( l_ode, x(t), t=0..1, [[x(0)=1],[x(0)=0],[x(0)=-1]], x=-2..2, arrows=SMALL, linecolor=[green,red,green], stepsize=0.1, title="Figure 3.6" );An equilibrium solution can also be called semi-stable if, as NiMlInRH increases, nearby solutions on one side approach the equilibrium, but nearby solutions on the other move away from the equilibrium. Adifferential equation with a semi-stable equilibrium would be the equationSS_ode := diff(x(t),t) = x(t)*(1-x(t))^2;The equilibrium solutions are NiMvJSJ4RyIiIQ== and NiMvJSJ4RyIiIg==, with the latter being the semi-stable equilibrium, as shown in Figure 3.7.DEplot(SS_ode, x(t),t=0..1, [[x(0)=1],[x(0)=.5],[x(0)=1.5]], x=0..2, arrows=medium, dirgrid=[10,10], linecolor=[red,green,green], stepsize=0.1, title="Figure 3.7" );Solutions starting above NiMvJSJ4RyIiIg== move away from this equilibrium, but solutions starting below NiMvJSJ4RyIiIg== move towards it. Solutions on one side of the equilibrium are attracted to the equilibrium, but solutions on the other side are repelled.

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