03 - Continuous Models

Discrete-Time Model

Let $N(t)$ denote the number of individuals in population at time $t$ then

$$\frac{dN}{dt}=f(N)$$

where $f$ is some function

Exponential Model

Models population where

1. New individuals birth rate is $b$

2. Individual death rate is $d$

Let

$$\frac{dN}{dt}=(b-d)N=rN$$

with solution

$$N(t)=N_0{e}^{rt}$$

where $N_0$ is initial population size at $t=0$

Logistic Model

Models Population with a Carrying Capacity so population size won’t exceed a limit

Let $K$ be the carrying capacity of the model then

$$\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$$

where $r>0$ and $K>0$

with solution

$$N=\frac{N_0}{\frac{N_0}{K}+\left(1-\frac{N_0}{K}\right)e^{-rt}}$$

Proof

Separation of Variables

$$\begin{array}{rl}\int \frac{1}{rN\left(1-\frac{N}{K}\right)}dN& =\int dt\\ \frac{1}{r}\int \frac{1}{N}+\frac{1}{K-N}dN& =\int dt\\ \frac{1}{r}(\ln (N)-\ln (K-N))& =t+C\end{array}$$

As $N(0)=N_0$ then

$$\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$$

Spruce Budworm Predation Model

Model population for a pest feeding on trees which kills them

$$\frac{dN}{dt}=r_{B}N\left(1-\frac{N}{K_B}\right)-\frac{B{N}^2}{A^2+N^2}$$

where $A,B>0$ with carrying capacity $K_B$ and growth rate $r_B$