1. CEE262C Lecture 3: Predator-prey models1 CEE262C Lecture 3: The predator- prey problem Overview Lotka-Volterra predator-prey model –Phase-plane analysis –Analytical solutions –Numerical solutions References: Mooney & Swift, Ch 5.2-5.3;

2. CEE262C Lecture 3: Predator-prey models2 Compartmental Analysis Tool to graphically set up an ODE-based model –Example: Population Immigration: ix Births: bx Deaths: dx Emigration: ex Population: x

3. CEE262C Lecture 3: Predator-prey models3 Logistic equation Population: x Can flow both directions but the direction shown is defined as positive

4. CEE262C Lecture 3: Predator-prey models4 Income class model Lower x Middle y Upper z

5. CEE262C Lecture 3: Predator-prey models5 For a system the fixed points are given by the Null space of the matrix A. For the income class model:

6. CEE262C Lecture 3: Predator-prey models6 Classical Predator-Prey Model Predator yPrey x Die-off in absence of prey dy Growth in absence of predators ax bxy cxy Lotka-Volterra predator-prey equations

7. CEE262C Lecture 3: Predator-prey models7 Assumptions about the interaction term xy xy = interaction; bxy: b = likelihood that it results in a prey death; cxy: c = likelihood that it leads to predator success. An "interaction" results when prey moves into predator territory. Animals reside in a fixed region (an infinite region would not affect number of interactions). Predators never become satiated.

8. CEE262C Lecture 3: Predator-prey models8 Phase-plane analysis

9. CEE262C Lecture 3: Predator-prey models9

10. 10 Analytical solution

11. CEE262C Lecture 3: Predator-prey models11 Solution with Matlab % Initial condition is a low predator population with % a fixed-point prey population. X0 = [x0,.25*y0]'; % Decrease the relative tolerance opts = odeset('reltol',1e-4); [t,X]=ode23(@pprey,[0 tmax],X0,opts); lvdemo.m function Xdot = pprey(t,X) % Constants are set in lvdemo.m (the calling function) global a b c d % Must return a column vector Xdot = zeros(2,1); % dx/dt=Xdot(1), dy/dt=Xdot(2) Xdot(1) = a*X(1)-b*X(1)*X(2); Xdot(2) = c*X(1)*X(2)-d*X(2); pprey.m

12. CEE262C Lecture 3: Predator-prey models12 at t=0, x=20 y=19.25

13. CEE262C Lecture 3: Predator-prey models13 Nonlinear Linear