# The Predator-Prey Equation

## Original Lotka-Volterra Model

We assume we have two species, herbivores with population x, and predators with propulation y. We assume that x grows exponentially in the absence of predators, and that y decays exponentially in the absence of prey. Consider, say, the system

## Critical points:

syms x y
vars = [x, y];
eqs = [x*(1-y/2), y*(-1+x/2)];
[xc, yc] = solve(eqs(1), eqs(2));
[xc, yc]
A = jacobian(eqs, vars);
disp('Matrix of linearized system:')
subs(A, vars, [xc(1), yc(1)])
disp('eigenvalues:')
eig(ans)
disp('Matrix of linearized system:')
subs(A, vars, [xc(2), yc(2)])
disp('eigenvalues:')
eig(double(ans))


ans =

[ 0, 0]
[ 2, 2]

Matrix of linearized system:

ans =

[  1,  0]
[  0, -1]

eigenvalues:

ans =

1
-1

Matrix of linearized system:

ans =

[  0, -1]
[  1,  0]

eigenvalues:

ans =

0 + 1.0000i
0 - 1.0000i



Thus we have a center at (2, 2) and a saddle point at (0, 0), at least for the linearized system. This suggests the possibility of periodic orbits centered around (2, 2).

## Phase Plot

rhs1 = @(t, x) ...
[x(1)*(1-.5*x(2)); x(2)*(-1+.5*x(1))];
options = odeset('RelTol', 1e-6);
figure, hold on
for x0 = 0:.2:2
[t, x] = ode45(rhs1, [0, 10], [x0;2]);
plot(x(:,1), x(:,2))
end, hold off


## Plot of Populations vs. Time

We color-code the plots so you can see which ones go together.

colors = 'rgbyc';
figure, hold on
for x0 = 0:10
[t, x] = ode45(rhs1, [0, 25], [x0/5; 2], options);
subplot(2, 1, 1), hold on
plot(t, x(:,1), colors(mod(x0,5)+1))
subplot(2, 1, 2), hold on
plot(t, x(:, 2), colors(mod(x0,5)+1))
hold on
end
subplot(2, 1, 1)
xlabel t
ylabel 'x = prey'
subplot(2, 1, 2)
xlabel t
ylabel 'y = predators'
hold off


## Modified Model with "Limits to Growth" for Prey (in Absence of Predators)

In the original equation, the population of prey increases indefinitely in the absence of predators. This is unrealistic, since they will eventually run out of food, so let's add another term limiting growth and change the system to

## Critical points:

syms x y
vars = [x, y];
eqs = [x*(1-y/2-x/4), y*(-1+x/2)];
[xc, yc] = solve(eqs(1), eqs(2));
[xc, yc]
A = jacobian(eqs, vars);
disp('Matrix of linearized system:')
subs(A, vars, [xc(1), yc(1)])
disp('eigenvalues:')
eig(ans)
disp('Matrix of linearized system:')
subs(A, vars, [xc(2), yc(2)])
disp('eigenvalues:')
eig(ans)
disp('Matrix of linearized system:')
subs(A, vars, [xc(3), yc(3)])
disp('eigenvalues:')
eig(double(ans))


ans =

[ 0, 0]
[ 4, 0]
[ 2, 1]

Matrix of linearized system:

ans =

[  1,  0]
[  0, -1]

eigenvalues:

ans =

1
-1

Matrix of linearized system:

ans =

[ -1, -2]
[  0,  1]

eigenvalues:

ans =

-1
1

Matrix of linearized system:

ans =

[ -1/2,   -1]
[  1/2,    0]

eigenvalues:

ans =

-0.2500 + 0.6614i
-0.2500 - 0.6614i



Thus we have saddles at (0, 0), (4, 0) and a stable spiral point at (2, 1).

rhs2 = @(t, x) ...
[x(1)*(1-.5*x(2)-0.25*x(1)); x(2)*(-1+.5*x(1))];
figure, hold on
for x0 = 0:.2:2
[t, x] = ode45(rhs2, [0, 10], [x0;1]);
plot(x(:,1), x(:,2))
end
for x0 = 0:.2:2
[t, x] = ode45(rhs2, [0, -10], [x0;1]);
plot(x(:,1), x(:,2))
end
axis([0, 4, 0, 4]), hold off

Warning: Failure at t=-3.380660e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (7.105427e-015) at time t.
Warning: Failure at t=-3.535072e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (7.105427e-015) at time t.
Warning: Failure at t=-3.735844e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (7.105427e-015) at time t.
Warning: Failure at t=-3.984664e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (7.105427e-015) at time t.
Warning: Failure at t=-4.299922e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.421085e-014) at time t.
Warning: Failure at t=-4.719481e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.421085e-014) at time t.
Warning: Failure at t=-5.332082e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.421085e-014) at time t.
Warning: Failure at t=-6.437607e+000.  Unable to meet integration tolerances without reducing the step size below the smallest value allowed (1.421085e-014) at time t.


## Plot of Populations vs. Time

figure, hold on
for x0 = 0:20
[t, x] = ode45(rhs2, [0, 25], [x0/5; 1], options);
subplot(2, 1, 1), hold on
plot(t, x(:,1), colors(mod(x0,5)+1))
subplot(2, 1, 2), hold on
plot(t, x(:, 2), colors(mod(x0,5)+1))
hold on
end
subplot(2, 1, 1)
xlabel t
ylabel 'x = prey'
subplot(2, 1, 2)
xlabel t
ylabel 'y = predators'
hold off