Bio 481 - Problem Set 4 - Dynamics 1

General Note: a^b stands for a raised to the b power.

1. Let's start looking into dynamics by taking the simplest equation we know, the exponential map (N(t+1)=r*N(t)). Assume r=1.5. Plot this map on a N(t+1) vs. N(t) graph. Also plot the N(t+1)=N(t) line. Are there any equilibria? Stair-step the population for a little bit to convince yourself that you understand the dynamics from this perspect.
   
   
2. Repeat problem 1, but for r=.85. Has anything changed? Modify the model to include a constant factor of migration, m=0.3 (N(t+1)=r*N(t)+m). Here make sure you stair-step the population from a couple of different starting locations.
   
   
3. Repeat problem 1 again, but for r=1.5. Also modify the model to include a constant factor of predation, p=0.2 (N(t+1)=r*N(t)-p). Here make sure you stair-step the population from a couple of different starting locations.
   
   
4. Now let us try the logistic map (N(t+1)=r*N*(1-N(t)/K)). Assume that K=1. Graph N(t+1) versus N(t) for several different r values. Make sure you also plot the line N(t+1)=N(t). Can you tell from the overall shape of the curve what the dynamics will be? Does the value of K change the dynamics? What do oscillatory attractors look like when you stair-step them?



5. Now modify the logistic map to include a constant predation factor (p=.2). Again graph N(t+1) versus N(t) for several different r values. How many equilbria are present here? Solve analytically for the equilibria. What is the dynamics of this map in the three regions of the curve?



6. Try a more complicated model. Here N0 represents offspring and N1 represents adults. Use the following two equations to set up your population: N0(t+1)=f*N1(t)*(1-N1(t)). N1(t+1)=s*N0(t). Do some time series projections so that you can find some starting conditions and parameter values where this population can persist. Now you want to set a graph something like N1(t+1) versus N1(t). Here we will use N1(t+1) versus N1(t). Why do we use this? Analytically find the equation used here. Analyze the dynamics.
   
   
7. Repeat the previous question, but now make the survivorship also density dependent. Use the form N1(t+1)=s*N0(t)*(1-N0(t)). Make sure you spend a lot of time with the parameter values here so you can look at regions with interesting behavior.
   
   

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