Biological populations with nonoverlapping generations?

In Robert May's work [1], he discussed the behavior of biological populations with non-overlapping generations. He showed that even simple, fully deterministic nonlinear models exhibit arbitrary behaviors ranging from stable equilibrium to stable cyclic oscillations and up to a chaotic regime depending on the range of values of the growth rate. 

One example of these simple nonlinear one-species models is given by equation, Eq. 1,  [1] which is considered by some to be the difference equation analog of the logistic differential equation where r is now the usual growth rate and K is the carrying capacity. 

$$N_{t+1} = N_t\bigl[r(1-\exp(N_t/K))\bigr]$$

Figure 1 shows the behavior of the population density $\frac{N_t}{K}$ as a function of time t as described by Eq.1, across different values of the growth rate r . For r = 1.8 the plot for the population density flattens after some time which suggests that the populations has reached a stable equilibrium point. For r = 2.3, the population density oscillates between two population points while for r = 2.6 the population density oscillates between four population points. This means that for r = 2.3 and r = 2.6, the population enters a 2-point and 4-points stable cyclic oscillation. For larger values of r, the population goes into a chaotic regime. Note that for the same value of r (r = 3.3) but different values of the initial population density (A: $\frac{N_0}{K}$ = 0.075, B: $\frac{N_0}{K}$ = 1.50), the population density has a tendency of being locked in a 3-point cycle as observed by May [1] in his results.  For even larger values or r (e.g. r = 4.0), the population variation becomes more extreme with periods of high population between periods of very low population. 

Figure 1. Dynamical behavior of the population density N/K as a function of time, t, 
as described by the difference equation (Eq. 1)  

Robert May [1] also found similar dynamical behaviors for the simple difference equation models for two species competition shown below:

$$N_1(t+1) = N_1(t) \exp \bigl[r_1(K_1 - \alpha_{11}N_1 - \alpha_{12}N_2)/K_1\bigr]$$

$$N_2(t+1) = N_1(t) \exp \bigl[r_2(K_2 - \alpha_{21}N_1 - \alpha_{22}N_2)/K_2\bigr]$$

where $r_i$ are the growth rates, $K_i$ are the carrying capacities and $\alpha_{ij}$ are the competition coefficients. 

The stability character of this difference equation model is shown in Figure 2. The first population $N_1/K$ is plotted as a function of time for several values of r. For the parameters, note that $r_1$ = r and $r_2$ = 2r ,  $K_1$ = $K_2$ = $K$, $\alpha_{11} = \alpha_{22} = 1$ and $\alpha_{12} = \alpha_{21} = 0.5$.

Figure 2. Stability character of the difference equation model for
two competing species described by Eq. 5.

Reference:

1. 1. May R. M. 

 1974 Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos.Science 186, 645–647. (doi:10.1126/science.186.4164.645)