The dynamics of a biological system can be described using differential equations. One example of biological system is the predator-prey system, which consists in two interacting species, the predators and the prey.
The population change in time of a predator-prey system can be mathematically modeled with the Lotka-Voltera equations. These are a set of first-order, nonlinear, differential equations:
\[ \begin{equation*} \begin{split}\frac{dx}{dt}=Ax – Bxy\\
\frac{dy}{dt}=Dxy – Cy
\end{split} \end{equation*} \]
where:
x – is the number of prey (e.g. rabbits)
y – is the number of predators (e.g. foxes)
dx/dt and dy/dt – are the growth rate in time of the population
t – time
A – the growth rate of prey in the absence of predators
B – the rate at which predators consume prey
C – death rate of predator in the absence of prey
D – rate at which the predators increase by consuming prey
The coefficients A, B, C and D are positive real parameters describing the interaction of the two species.
The Xcos block diagram implementation of the predator-prey system (Lotka-Voltera equations) is the following:
Image: Predator-prey model – Xcos block diagram
As you can see, we integrate the right side of differential equations using an INTEGRAL_f block. The outputs of the integrator blocks represent the variation in time of the population of prey x(t) and predators y(t). The coefficients A, B, C and D are set as parameters of the GAINBLK blocks.
As initial conditions we assume that the prey population is 6 and the predator population is 2.
Image: Predator-prey model – simulation parameters
The simulation is set to run for 12 seconds, with the default Xcos settings for integration method and solver tolerance. The coefficients A, B, C and D are set using the Set Context menu dialog.
Image: Predator-prey model – constant parameters setup
The population number in time, for both prey and predators, are saved in the Scilab workspace as structure variables: x_prey is the prey population number in time x(t), and y_predator is the predator population number in time y(t).
We have a post-processing Scilab script which will plot the number of prey and predator population versus time.
plot(x_prey.time,x_prey.values,'b','Linewidth',2)
plot(y_predator.time,y_predator.values,'r','Linewidth',2)
legend("prey","predator",1)
hf=gcf();
hf.background = 8;
ha=gca();
ha.grid = [1,1]
ha.children(1).line_mode="off";
xlabel("time",'fontsize',2)
ylabel("x, y",'fontsize',2)
title("Predator-prey population growth in time",'fontsize',2) By running the above Scilab script we get the plot of population number in time, for both prey and predators.
Image: Predator-prey population number in time
Notice that the population number at time 0 s starts with the initial values set in the integrator blocks, 6 for prey and 2 for predators.
By analyzing the plot of the population number, we can notice that:
- the predator population is always behind the prey population
- when the prey population is high, the predators start to increase due to abundance of food
- when the predators are too many, the prey population starts to decline
- when the prey is not anymore sufficient, the predators population starts to decline due to lack of food
- when the predators population is low, the prey population increases due to low mortality rate
For a different view of the results we can plot the population number of the predators y(t) versus the population number of the prey x(t) with the following Scilab script:
idx=find(x_prey.values < 6.01 & x_prey.values > 5.99);
plot(x_prey.values(1:idx(2)),y_predator.values(1:idx(2)),'Linewidth',2)
plot(x_prey.values(1),y_predator.values(1),'s')
xgrid
xlabel("Prey, x",'fontsize',2)
ylabel("Predator, y",'fontsize',2)
title("Predator vs. prey population growth",'fontsize',2) By running the above Scilab instructions, we get the following plot:
Image: Predator vs. prey population number
The same analysis can be done on the vs. plot above. The start point are the initial values, 6 prey and 2 predators. Following the contour of the plot, we can notice the same behavior as in the time plot. With the initial population number, the predators start to increase, due to abundance of food, while the prey starts to decline.
The apex of the predator population is at 3.94, an the minimum is 0.36. The maximum number of the prey population is 6.13 and the minimum is 1.17.
The number of predators and prey should be regarded as relative numbers, for example, number of individuals on 10 km2.
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