(WHALES AND KRILL PROBLEM)

Here we will investigate the population dynamics of whales and krill, where krill is assumed to be the main source of food for the whales. We assume the following model: k' = (a-bw) k w' = (-m+nk) w where k is the krill population (as a function of the time in years), w is the whale population, and a, b, m and n are positive constants.

Interpret this system of differential equations and suggest the meaning of the constants. (HINT: it can be difficult to give precise meanings to the constants. but look at the terms and try to interpret them in terms of what they model in reality.)

Determine if there are any equilibrium points of this system. If so, can you say anything about the stability of these points?

Write the equations for simulating this system with Eulers method. Consider the parameter values a=0.2, b=0.0001, m=0.5 , n=0.000001. With the starting points k(0)= 700000 and w(0)=3000, and time step deltaT= 0.3 years, manually simulate k(t) and w(t) for t=0.3 and t=0.6.

Use the parameters given in c) and use Mathematica to simulate the model from several different starting points. To get interesting results do not just pick arbitrary starting points, but be systematic and think how they relate to the result in b). Discuss your observations. Can you draw general conclusions not dependent on this particular choice of values? You can use the Mathematica function NDSolve for the simulation of k(t) and w(t). You are recommended to first plot and study the two functions k(t) and w(t) (with the function Plot), before you plot with the function ParametricPlot. Here is some Mathematica code to help you get started: a=0.2 b=0.0001 m=0.5 n=0.000001 s=NDSolve[{k'[t]==(a-b*w[t])k[t],w'[t]==(-m+n*k[t])w[t], k[0]==70000,w[0]==3000},{k,w},{t,1000}] Plot[Evaluate[{k[t]}/.s],{t,0,100}] Plot[Evaluate[{w[t]}/.s],{t,0,100},PlotRange->All] ParametricPlot[Evaluate[{k[t],w[t]}/.s],{t,0,100},AspectRatio->1,PlotRange->All]

Investigate the effect of krill fishing on these populations. To model this, we add a term -rk to the equation for k', where r

Can you suggest how the model could be refined or extended?