If a population is growing in a constrained environment with carrying capacity [latex]K[/latex], and absent constraint would grow exponentially with growth rate [latex]r[/latex], then the population behavior can be described by the logistic growth model:
where [latex]t[/latex] stands for time in years, [latex]c[/latex] is the carrying capacity (the maximal population), [latex]P_0[/latex] represents the starting quantity, and [latex]r[/latex] is the rate of growth.
For now we will use the model in the form [latex]{{P}_{n}}={{P}_{n-1}}+r\left(1-\frac{{{P}_{n-1}}}{K}\right){{P}_{n-1}}[/latex] but it is important to know both forms of the model.
Unlike linear and exponential growth, logistic growth behaves differently if the populations grow steadily throughout the year or if they have one breeding time per year. The recursive formula provided above models generational growth, where there is one breeding time per year (or, at least a finite number); there is no explicit formula for this type of logistic growth.
A forest is currently home to a population of [latex]200[/latex] rabbits. The forest is estimated to be able to sustain a population of [latex]2000[/latex] rabbits. Absent any restrictions, the rabbits would grow by [latex]50\%[/latex] per year. Predict the future population using the logistic growth model.
Modeling this with a logistic growth model, [latex]r = 0.50[/latex], [latex]K = 2000[/latex], and [latex]P_0 = 200[/latex]. Calculating the next year:
A calculator was used to compute several more values:
[latex]n[/latex]
[latex]0[/latex]
[latex]1[/latex]
[latex]2[/latex]
[latex]3[/latex]
[latex]4[/latex]
[latex]5[/latex]
[latex]6[/latex]
[latex]7[/latex]
[latex]8[/latex]
[latex]9[/latex]
[latex]10[/latex]
[latex]P_n[/latex]
[latex]200[/latex]
[latex]290[/latex]
[latex]414[/latex]
[latex]578[/latex]
[latex]784[/latex]
[latex]1022[/latex]
[latex]1272[/latex]
[latex]1503[/latex]
[latex]1690[/latex]
[latex]1821[/latex]
[latex]1902[/latex]
Plotting these values, we can see that the population starts to increase faster and the graph curves upwards during the first few years, like exponential growth, but then the growth slows down as the population approaches the carrying capacity.
A field currently contains [latex]20[/latex] mint plants. Absent constraints, the number of plants would increase by [latex]70\%[/latex] each year, but the field can only support a maximum population of [latex]300[/latex] plants. Use the logistic model to predict the population in the next three years.
