Writing and Analyzing Population and Predator-Prey Models

translate into change equations population models with or without logistic growth ▶ translate into change equations verbal descriptions of dynamical systems involving interactions. This implies: ▶ knowing how to write down the term representing an interaction.

Population model Basics 

• x(t): nb of individuals at time t

• x’(t): rate at which the population changes at time t

• per capita= “per head” 

  • ex. rabbits give birth to babies w a per capita birthrate b → b X(t)

  • death rate per capita → d X(t)

  • This born and die model is structured like this: X’(t)= b X(t) - d X(t)= (b-d) X(t)=r X(t)

• population of # animals → x(t)=#

Including resource limitations into the basic model population equation

• There is such a thing as exponential growth, but that’s unrealistic, as there is no such thing as unlimited resources 

• Therefore, there should be a limit of resources, and with that comes logarithmic growth 

• Introduces the idea of a carrying capacity, k. 

• X/k represents the fraction of resources used by the population

• (1-X/k) represents the fraction of available resources

• The Logistic Model: X’(t) = r X(t) (1- X(t)/k)

  • This represents how population growth slows as resources become limited, eventually stabilizing at the environment’s carrying capacity.

  • grows to a saturation, a carrying point,

Lotka-Volterra Model (aka Shark-Tuna Model) 

Goal: model the evolution of predator and prey populations

Equation for Preys.

1. Nb. of predators denoted S(t), nb. of preys denoted T(t) 

2. Without predators, prey follow exponential growth (for natural births and deaths)

• b T(t)

3. Predators eat prey. Assume that encounters happen at random. 

• S(t) T(t)

4. Only some encounters lead to predators eating the prey! 

• B for beta

• B S(t) T(t)

T’(t) = b T(t) - B S(t) T(t)

Equations for Predator.

1. The predator only feeds itself with this type of prey

2. Number of predator babies is directly proportional to the number of prey each predator eats

• In other words, births of predators per unit of time

• m = proportionality constant

• m B S(t) T(t)

3. predators die at a per capita rate of d

• d S(t)

S’(t) = m B S(t)T(t) - dS(t)

Three-Species Model