USC3002 Picturing the World Through Mathematics Wayne Lawton Department of Mathematics S , Theme for Semester I, 2008/09 : The Logic of Evolution, Mathematical Models of Adaptation from Darwin to Dawkins

Differential Equations Ordinary : if they involve one or more functions of a single independent variable, otherwise they are Partial Ordinary: exponential growth;circular motion Partial: heat equation

Differential Equations Solution Equation Linear Autonomous Properties Linear Nonaut. Nonlinear Autonomous Autonomous:depends only on Nonlinear Autonomous

System of 1 st Order ODE’s Definition A solution is a map of function such that Theorem (Cauchy-Peano) Theorem (Picard–Lindelöf)

Circle Equations Exact solutions for Remarks Linear, existence of closed form solutions is useful for theory and for testing the accuracy of numerical solutions

Lotka-Volterra Equations Exact solutions for exist but they do not have simple closed forms. Therefore numerical solutions are useful. where

Lotka-Volterra Equations Exact solutions for exist but they do not have simple closed forms. Therefore numerical solutions are useful. where

LV Equations: Numerical Solution function [x,y,t] = circle(x0,y0,dt,T,a,b,c,d) %function [x,y,t] = circle(x0,y0,dt,T,a,b,c,d) % % Your Name, Date % Computes solution of the LV % Predator-Prey Equations % dx/dt = x(a-by) % dy/dt = x(-c+dx) % Inputs: (x0,y0) = initial point % dt = time increment % t = total time % Outputs: x,y arrays containing values % of the solutions at times t = 0:dt:T % x(1) = x0; y(1) = y0; n = 2; t = dt; while t <= T x(n) = x(n-1) + x(n-1)*(a - b*y(n-1))*dt; y(n) = y(n-1) + y(n-1)*(-c + d*x(n-1))*dt; t = t + dt; n = n + 1; end t=0:dt:T; first half of programsecond half of program

LV Equations: Numerical Solution >> x0 = 2; y0 = 1; dt = 0.001; T = 4*pi/3; >> a = 1; b = 1; c = 1; d = 1; [x,y,t] = circle(x0,y0,dt,T,a,b,c,d); plot(x,y) >> grid

LV Equations: Some Solutions equilibrium

LV Equations: Periodicity equilibrium If is a solution of the LV Eqns so V is constant on the solution curves. Since the curves V = constant are are closed the solutions curves are periodic.

Volterra’s Principle Fishing reduces a and increases c, WWI reduced fishing and therefore increased a and decreased c. This increased/decreased the equilibium (and hence average) numbers of predatory/prey fish during WWI.

LV Equations with Intraspecific Competition where If isoclines in interior of positive quadrant do not intersect they divide it into 3 regions and all solutions to (a/e,0). If isoclines intersect at an equilibrium point, then they divide the interior into 4 regions. Lyapunov  solutions converge to eq.point.

Homework 6. Due Monday Use the computer program that you used to generate solutions of the LV Equations during last weeks lab to compute the solutions of several solutions for each set of LV Equations with intraspecific competition that correspond to the parameters (a,b,c,d,e,f) that you found in For the LV Equations with intraspecific competition find a set of values for (a,b,c,d,e,f) for which the isoclines do not intersect (in the 1 st = positive quadrant) and another set of values for which they do intersect (in the 1 st quadrant). Then compute the equilibrium points for each of these two cases.