10.5.1: Logistic population growth

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INTRODUCTION

This material in this chapter has been adapted from Donovan and Welden (2002).

Donovan, T. M. and C. Welden. 2002. Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA.

As in the previous section on Geometric and Exponential Growth, we begin with a model of population dynamics in discrete time, with explicit parameters for per capita rates of birth and death.

In the previous section, we developed the following geometric model of population dynamics:

\[N_{t+1}=N_{t} + b*N_{t} - d*N_{t} \nonumber\]

where

\(N_{t} \nonumber\) =population at time \(t\)

\(N_{t+1} \nonumber\) = population at one time unit later

\(b \nonumber\)= per capita birth rate

\(d \nonumber\) = per capita birth rate

As you discovered in the earlier exercise, this model produces geometric population growth (the discrete-time analog of exponential growth) if \(b\) and \(d\) are held constant and \(b > d\) However, the assumption that per capita rates of birth and death remain constant is unrealistic, so we will now develop a model in which these rates change. Specifically, we will consider only one cause of changes in per capita birth and death rates: the size of the population itself. In other words, we will assume that environmental conditions, food supply, and so on remain constant; only the size of the population itself changes. These additions result in the logistic growth model.

Because per capita rates of birth and death do change in response to population size or density, logistic models are density-dependent, in contrast to geometric and exponential models, which are density-independent. As the population grows, less food and water, fewer nesting and hiding sites, and fewer resources in general are available to each individual, affecting both an individual’s rate of reproduction and its risk of death. Our model will thus include intraspecific competition (competition among members of the same species) for resources. These models are used to inform practical decisions in the management of fisheries and game animal populations and are used to predict the growth of the human population. Later exercises will develop models of interspecific (between two species) competition and predator-prey dynamics.

Logistic growth models include an equilibrium population size in this model. In other words, populations grow until they reach a stable size. The population is at equilibrium when total deaths equal total births and when per capita rates of birth and death are equal. This equilibrium populations size is so important in population biology, it is given its own name—the carrying capacity. The carrying capacity is defined as the largest population that can be supported indefinitely, given the resources available in the environment. This carrying capacity is represented by the parameter \(K\).

\[\frac{dN}{dt} = rN \frac{(K-N_{t})}{K} \nonumber\]

If we begin with a very small population, the term \(\frac{(K-N_{t})}{K} \) is very nearly equal to \(\frac{(K)}{K} \) or 1. The model will then behave like a geometric model, and the population will grow, provided \(r > 1 \). The population will grow slowly at first, because the parameter \(r\) is also being multiplied by a number \(N_{t}\) that is nearly equal to zero, but it will grow faster and faster, at least for a while. At some point, however, population growth will begin to slow because the term \(\frac{(K-N_{t})}{K}\) is getting smaller and smaller as \(N_{t}\) gets larger and closer to \(K\).

At the other extreme, imagine a population that starts out at a size very close to its carrying capacity, K. The term \(\frac{(K-N_{t})}{K}\) becomes nearly equal to zero, and population growth is extremely slow. When \(N_{t} = K\) , the population stops growing altogether.

EXPLORE THIS MODEL

Before moving on to the next section, explore the Logistic growth Shiny App developed by Dr. Aaron Howard to better understand how changes to the initial population size \(N\), carrying capacity \(K\), and the population growth rate \(r \) impact population size over time.

References

Donovan, T. M. and C. Welden. 2002. Spreadsheet exercises in ecology and evolution. Sinauer Associates, Inc. Sunderland, MA, USA.