BIO 360  Ecology

Lecture 23: Predator-prey theory

I.  Predator-prey oscillations:  a model

    A.  The Lotka-Volterra model of predator and prey population dynamics (p. 325-328 in your book)

    B.  Two equations, one for predator (Population size denoted as P for predator instead of N) and one for prey (N)

    C.  Without the predator, assume exponential population growth of prey, dN/dt = rN
        1.  Assumption 1: prey population growth is only limited by predators
        2.  Assumption 2:  predator is a specialist, consuming only this prey
        3.  Assumption 3:  Predator population can consume an infinite number of prey

       

        4.  With the predator, assume that population is reduced by the predator in proportion to the predator's capture rate, c
            -- thus, dN/dt = rN - cNP
 

    D.  Without prey, assume that predator population sizes decline exponentially, dP/dt = -dP
        1.  d is the death rate
 

        2.  In the presence of the prey, assume that predator population growth depends on the capture rate of prey, and on the rate at which predators convert prey to energy
            -- thus, dP/dt = acNP - dP

       
 
 

    E.  Use a graphical approach to solving for the dynamics of predator and prey population sizes

        1.  Solve Lotka-Volterra equations  one species at a time for solutions at equilibrium (when population size is not changing)
 
 
 

        2.  Graph the line represented by the equilibrium solutions
 
 
 
 
 

        3.  Determine population growth from any initial population size
 
 
 
 
 

II.  Changing the assumptions of the initial model to make it more realistic

    A.  Relax Assumption 1:
        -Prey controlled by predator and by other density-dependent factors (like limits on food resources)

        -Implications: Stabilizes interactions (no oscillations)

    B. Relax Assumption 1:
        - In addition to above, prey population growth limited at low prey population sizes
        Might be hard to find mates

        - Implications: Stability depends on predator efficiency.  If they are efficient, then predators will hunt prey to extinction. 
 
    C. Relax assumption 3:
        - Predators are not able to consume infinite prey
        -Functional response of predators to prey density

        - Implications: Predators that switch prey or that reach carrying capacity stabilize predator-prey interactions

        - Final note: if realistic assumptions tend to stabilize dynamics, what causes Oscillations? 
                    - Time delays