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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 \(N_{t+1} = N_t + Births - Deaths = N_t +b N_t - d N_t = N_t (1 + b - d) = R N_t \). So, \(R ∈ [0, b_{max} + 1]\).

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.
\(R < 1 \implies b < d\) and \(N_{t} → 0\) as \(t → \infty\).
For R = 1, b=d and \(N_{t+1} = N_t\)
\(R > 1 \implies N_{t} → \infty\) as \(t → \infty\).

R linearly reducing with population

Model: \(R = R_0 (1-\dfrac{N_t}{k})\) for \(N_t \leq k\), leading to \(N_{t+1} = R_0 N_t(1-\dfrac{N_t}{k})\). 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 \(x ∈ [0, 1]\)
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.