转载自：http://www-users.math.umd.edu/~jmr/246/predprey.html
The Predator-Prey Equation
Contents
- Original Lotka-Volterra Model
- Critical points:
- Phase Plot
- Plot of Populations vs. Time
- Modified Model with "Limits to Growth" for Prey (in Absence of Predators)
- Critical points:
- Plot of Populations vs. Time
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
Published with MATLAB® 7.0