Discrete Time

General framework

Let $N_{t}$ be reproducing individuals at time $t$ . Let $b$ be offspring per individual per generation, who survive to become reproducing adults. Let d = death probability per individual per generation. Then, number of individuals one generation hence is $N_{t + 1} = N_{t} + B i r t h s - D e a t h s = N_{t} + b N_{t} - d N_{t} = N_{t} (1 + b - d) = R N_{t}$ . So, $R \in [0, b_{m a x} + 1]$ .

Oft, R is a function of $N$ .
Often, $d = 1$ - non-overlapping generations.

Constant R cases

Density independent population growth.
So, $N_{t + 1} = R^{t} N_{1}$ . Exponential growth.
$R < 1 ⟹ b < d$ and $N_{t} \to 0$ as $t \to \infty$ .
For R = 1, b=d and $N_{t + 1} = N_{t}$
$R > 1 ⟹ N_{t} \to \infty$ as $t \to \infty$ .

R linearly reducing with population

Model: $R = R_{0} (1 - \frac{N_{t}}{k})$ for $N_{t} \leq k$ , leading to $N_{t + 1} = R_{0} N_{t} (1 - \frac{N_{t}}{k})$ . Taking $x_{t} = N_{t} / k$ (aka rescaled population to get “population density”), this can be simplified to $x_{t + 1} = R_{0} x_{t} (1 - x_{t})$ for $x \in [0, 1]$
Logistic population growth. Population converges to some finite value.
Equilibria: $x_{t+1} = x_t = x^$. So $x^= R_0(1-x^*)$
- $x^{*} = 0$ for all $R_{0}$
- $x^{*} = (R_{0} - 1) / R_{0}$ for all $R_{0} > 1$
- So, 1 is a bifurcation point, where stability of the population changes.
$N_{t}$ shows oscillations about the equilibrium - Entirely because of internal dynamics of the population.

R exponentially decays

Model: $R = e^{r (1 - \frac{N_{t}}{k})}$ , with $N_{t + 1} = R N_{t}$ . $e^{r}$ is the equivalent of $R_{0}$ , the max rate of growth. If k is $\infty$ , $N_{t}$ grows at max rate. If k is $N_{t}$ , numbers stay constant. So, $k$ is max numbers that the environment can support.