Question 10.11: The predator-prey problem. (Using MATLAB's built-in function......

Numerical Methods for Engineers and Scientists: An Introduction With Applications Using MATLAB [1013610]

The predator-prey problem. (Using MATLAB’s built-in function to solve a system of two first-order ODEs.)

The relationship between the population of lions (predators), $N_{L}$ , and the population of gazelles (prey), $N_{G}$ , that reside in the same area can be modeled by a system of two ODEs. Suppose a community consists of $N_{L}$ lions (predators) and $N_{G}$ gazelles (prey), with $b$ and $d$ representing the birth and death rates of the respective species. The rate of change (growth or decay) of the lion $(L)$ and gazelle populations can be modeled by the equations:

$\begin{aligned}& \frac{d N_{L}}{d t}=b_{L} N_{L} N_{G}-d_{L} N_{L} & (10.173)\\& \frac{d N_{G}}{d t}=b_{G} N_{G}-d_{G} N_{G} N_{L}\end{aligned}$

Determine the population of the lions and gazelles as a function of time from $t=0$ to $t=25$ years, if at $t=0, N_{G}=3000$ , and $N_{L}=500$ . The coefficients in the model are: $b_{G}=1.1 \mathrm{yr}^{-1}$ , $b_{L}=0.00025 \mathrm{yr}^{-1}, d_{G}=0.0005 \mathrm{yr}^{-1}$ , and $d_{L}=0.7 \mathrm{yr}^{-1}$ .

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Script File

In this problem $N_{L}$ and $N_{G}$ are the dependent variables, and $t$ is the independent variable. To solve the problem a user-defined function named PopRate $(t, N)$ is written. The function, listed below, calculates the values of $\frac{d N_{L}}{d t}$ and $\frac{d N_{G}}{d t}$ in Eq. (10.173). The input argument $\mathrm{N}$ is a vector of the dependent variables, where $N(1)=N_{L}$ and $N(2)=N_{G}$ . Notice how these components are used when the values of the differential equations are calculated. The output argument dNdt is a column vector where the first element is the value of the derivative $\frac{d N_{L}}{d t}$ , and the second is the value of the derivative $\frac{d N_{G}}{d t}$

function dNdt = PopRate(t, N)

bG = 1.1; bL = 0.00025 ; dG = 0.0005 ; dL = 0.7;

f1 = bL*N(1)*N(2) - dL*N(1) ;

f2 = bG*N(2) - dG*N(2)*N(1) ;

dNdt = [fl ; f2] ;

The system of ODEs in Eq. (10.173) is solved with MATLAB's solver ode45. The solver uses the user-defined function PopRate for calculating the right-hand sides of the differential equations. The following MATLAB program in a script file shows the details. Notice that $\mathrm{Nin} i$ is a vector in which the first element is the initial value of $N_{L}$ and the second element is the initial value of $N_{G}$ .

tspan = [0 25] ;

Nini= [500 3000] ;

[Time Pop] = ode4 5 ( @PopRa te , tspan, Nini) ;

plot(Time,Pop(:,1) ,'-' ,Time,Pop(:,2) ,'--')

xlabel('Time (yr)')

ylabel('Population')

legend('Lions' ,'Gazelles')

In the output arguments of the solver ode45 , the variable $\mathrm{T}$ ime is a column vector, and the variable Pop is a two-column array, with the solution of $N_{L}$ and $N_{G}$ in the first and second columns, respectively. The first few elements of [Time Pop] that are displayed in the Command Window (if the semicolon at the end of the command is removed) are shown below. The program generates a plot that shows the population of the lions and gazelles as a function of time.

Time =

0
0.0591
0.1182
0 .1773
…

Pop =

1.0e+003 *
0.5000 3.0000
0.5020 3.1545
0.5053 3.3166
…