Modeling - Math 056 - Homework 3

1. Indicate whether each of the following equations is linear or nonlinear. If linear, determine the solution; if nonlinear, find any steady states of the equation.

(a) x_n = (1 - α)x_n-1 + βx_n

(b) x_n+1 = x_ne^-ax_n

(c) x_n+1 = K/k₁+k₂/x_n

2. Find the steady states and determine when they are stable. Then sketch the functions f(x) given in this problem. Use the cobwebbing method to sketch the approximate behavior of solutions to the equations for some initial starting value of x₀.

(a) x_n+1 = 1/2+x_n

(b) x_n+1 = x_n ln x²_n

3. In population dynamics, a frequently encountered model for fish populations is based on an empirical equation called the Ricker equation:

N_n+1 = αN_ne^-βN_n.

In this equation, α represents the maximal growth rate of the organism and β is the inhibition of growth caused by overpopulation.

(a) Find the steady states.

(b) Determine when the steady states are stable.

(c) Biologically interpret the results of (b).

4. Consider the equation

N_t+1 = N_te^r(1-Nt/K).

The equation models a single-species population growing in an environment that has a carrying capacity k. By this, we mean that the environment can only sustain a maximal population level N = K. The expression

γ = e^r(1-Nt/K)

reflects a density dependence in the reproductive rate. To verify this observation consider the following steps:

(a) Sketch γ as a function of N_t.

(b) Find the steady states of the equation.

(c) Determine when the steady states are stable and explain the behavior for various parameter regimes.

(d) Using a calculator or simple computer program, plot successive population values N_t for some choice of parameters r and K.

5. The difference equation

N_t+1 = (r₁ + (r₂/1 + e^A(N_t-B)))N_t

is used for simulating the growth of two interacting populations. Find the steady states and determine when each is stable.

6. In class we discussed period 2 doubling. On the graph below, complete the cobwebbing for the sequence x_n+1 = rx_n(1 - xn) for r = 3.3. What do you notice about your map? Please use a ruler for accuracy.