## Discrete Generations

Nonoverlapping generations are common biological phenomenon and population growth is modeled using difference equations. These models all assume that the population size at time t +1 depends only on the conditions at the time t (usually the beginning of each growing or breeding season). The simplest model can be expressed as

where Nt +1 is the population size at time t + 1 and Nt is the population size at time t. R0 is a constant per capita reproductive rate, which is independent of population size. There are three possible outcomes of this model. If R0 > 1 the population will increase geometrically (A, Figure 1), if R0 = 1 the population is at equilibrium (B, Figure 1), and if R0 < 1 the population size will decrease (C, Figure 1).

The above model assumes that R0 does not change with population size; however, populations do not generally grow at a constant rate. The growth rate of a population should be different for different densities. The simplest assumption regarding this change in reproductive rate based on density is to assume that the R0 decreases linearly as population size increases. R0 can be calculated by using a linear equation:

 500 450 400 e 350 si 300 n io 250 13 200 a. o 150 CL 100 50 Time (generations) Figure 1 Population growth for discrete generations using a difference eqn [1] with N0 = 100. ArepresentsR0 = 1.1; B represents R0 = 1.0, and C represents R0 = 0.9. where (—)B is the slope of the line and Neq is the equilibrium size of the population (R0 = 1 at Neq). Therefore, eqn [1] can be rewritten as This model can generate several different behaviors, depending on the value of BNeq. If BNeq is between 0 and 2, the model will converge on Neq either without oscillations (1> BNeq>0) or with damped oscillations (2 > BNeq>1) 2 and 2.57, the growth ec (Figure 2a). If BNeq is between B = 0.018 0 10 20 30 40 50 60 70 80 90 100 Time (generations) Figure 2 Population growth with discrete generations and R0 as a linear function of population density (eqn [3]). The model parameters were N0 = 20, Neq = 150, B = 0.012 (a), B = 0.015 (b), and B = .018 (c). Modified from Krebs CJ (2001) Ecology, 5th edn. San Francisco, CA: Benjamin Cummings. form is stable limit cycles continuing indefinitely (Figure 2b), and if BNeq> 2.57, the population size fluctuates in a nonrepeatable (chaotic) pattern (Figure 2c).