Lecture 12 Population Models

Population Modeling

Changes to a population can be modeled as a "mass balance" relation:

  • Inputs: Births & Immigration
  • Outputs: Deaths and Emigration

Syntax for model:

  • N(t) = population at time t
  • $dN\over dt$ = instantaneous rate of population change
  • b(N) = per capita birth rate when the population has size N
  • d(N) = per capita death rate at population size N
  • r = intrinsic rate of growth = b(N)-d(N) when migration is negligible
  • K = carrying capacity

Equation Development

  • In reality, births and deaths do not provide continuous changes in the population. That is, each birth or death changes the population by an integer value, at a well-spaced time. Consequently, if we considered a true population as a function of time, it would in reality be comprised of flat steps with discrete jumps. However, when the population is sufficiently large, we can essentially average over these small fluctuations and consider the average behavior of the population.
  • Suppose b represents the per capita birth rate, and d represents the per capita death rate. Then the rate of births in a population of size N would be bN, and the rate of deaths would be dN. The differential equation of population growth can thus be developed:
(1)
\begin{align} \frac{dN}{dt} = b*N - d*N. \end{align}
  • When we aren’t concerned with keeping separate track of birth and death rates, we often combine them into a per capita growth rate, r = b − d. To relate this back to the idea of per capita growth, if we divide the total rate of growth by the number of individuals in the population, this measures the per capita growth:
(2)
\begin{align} \frac{1}{N}*\frac{dN}{dt} = b - d = r \end{align}
  • Although not outlined above, both the birth rate and the death rate could vary. In particular, birth and death might depend on the population size and the time: b = b(t,N) and d = d(t,N).
  • Population models are initial value problems, and so must identify a starting value, or initial condition, $N(0)=N_{o}$.
  • Intrinsic growth rate largely determines if a population will experience exponential growth (r > 0) or decay (r < 0): $N(t) = N_{o}{e}^{rt}$

Cases to examine:

Case 1: Constant per capita birth/death rates

  • Function of intrinsic growth rate and population
  • $\frac{dN}{dt} = r*N$
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Case 2: Crowding effects and logistic growth

  • Known as Verhulst model or logistic equation
  • Function of intrinsic growth rate, population, and carrying capacity
  • $\frac{dN}{dt} = r*N(1-\frac{N}{K})$
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Case 3: Allee Effect

  • Certain populations are sensitive to environmental or social factors, which can be modeled through added complexity (ex. a predator that requires a pack to hunt effectively, but is hindered in overcrowded conditions).
  • Function of intrinsic growth rate, population, carrying capacity, and conditional capacity, ${K}_{o}$
  • $\frac{dN}{dt} = r*N(\frac{N}{K_{o}}-1)(1-\frac{N}{K})$
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