WLF 448: Fish & Wildlife Population Ecology

Lab 14: Predation

I. Objectives

1. Review concepts of predator-prey interactions, including properties that affect stability.

2. Use programs POPEC and POPULUS to gain a better understanding of some mathematical models of predation (i.e., Lotka-Volterra and Tanner's models) and their predictions.

II. Definitions and Concepts

A. Definitions

• Predator: an organism that uses other live organisms (prey) as an energy source and, in doing so, removes the prey individuals from the population (Colinvaux 1986).

• Parasitoids: any number of so-called parasitic insects whose larvae live within and consume their host, usually another insect, inevitably leading to the death of the host (Ricklefs 1990).

• Parasite: an organism that consumes part of the blood or tissues of its host, usually without killing the host.

• Herbivore: an organism that consumes living plants or their parts; herbivores may function either as predators (consuming whole plants) or parasites (consuming living tissues but not killing their victims).

B. Questions to Ask When Studying Predator-Prey Interactions (C.J. Holling)

1. How would the prey population grow in the absence of the predator?

2. How many prey does each predator consume, i.e., what is the functional response?

3. How does predator density change when prey density changes, i.e., what is the numerical response?

We might also ask about the stability of the predator-prey interactions.

C. Functional Response and Stability of Predator-Prey Interaction

• Type 1 Functional Response – as prey increases, predator search time decreases so there is an initial rapid increase in prey numbers taken. However, there usually is some upper limit to the number of prey a single predator can handle per unit of time. A type I response is neutral until an upper limit is reached, then it tends to be destabilizing.

• Type 2 Functional Response – similar response to Type 1, but the upper limit is reached more slowly. Tends to be destabilizing.

• Type 3 Functional Response – sigmoid-shaped curve possibly due to prey switching, which can occur when the predator concentrates on the prey species with the largest densities. This causes a delay between when the prey begins to increase and when the predator detects the increase and switches effort. Tends to be stabilizing.

D. Numerical Response and Stability of Predator-Prey Interaction

• Type 1 Numerical Response – linear relationship where predator's K is set by prey density. Neutral in terms of stabilizing effect.

• Type 2 Numerical Response – increase in predator numbers with increased prey abundance, but predator population reaches a threshold where something else limits the population (e.g., territories, nest sites, etc.). Somewhat destabilizing.

• Type 3 Numerical Response – a density-dependent predator response to prey abundance (sigmoid-shaped curve). Stabilizing effect up to some threshold level.

• Type 4 Numerical Response – No predator response to changes in prey abundance; the number of predators per prey gets smaller. Destabilizing.

• Type 5 Numerical Response – negative response in predator abundance occurs as prey population increases. Destabilizing.

You need to look at functional and numerical responses together in order to gain insight into the stability of predator-prey interactions.

III. Lotka-Volterra Model

Population growth for prey population:

dH / dt = r H - b₁ H P

where,

H = number of prey (H for herbivores) 
P = number of predators 
r = intrinsic rate of growth for prey population 
b1 = predation rate (coefficient expressing the efficiency of predation). 

Population growth for predator population:

dP / dt = -m P + b₂ H P

where,

b2 = rate of growth of the predator per unit contact with prey
 m = predator's intrinsic rate of growth (decline) in absence of prey

The simultaneous solution of the Lotka-Volterra equations for predation predicts that numbers of both predators and prey should oscillate, and the oscillations should be coupled (Coupled Oscillation Hypothesis).

Numbers of both predators and prey circle a singular stable point in perpetually balanced imbalance, i.e., neutral limit cycle. The degree of oscillation about the equilibrium point is a function of initial population sizes. Perturbations may eventually result in extinction of prey and predator.

Assumptions

1. Unrestricted exponential growth of the prey population in the absence of predators.

2. Environment is homogeneous.

3. Every prey has an equal probability of being attacked, e.g., no age-based selection.

4. Predators have an unlimited capacity for increase, i.e., the response of predators to prey is linear (i.e., a Type 1 numerical response).

5. Prey density has no effect on the probability of being eaten, i.e., a Type 1 functional response.

6. Predator density has no effect on the probability of a predator capturing prey.

7. No time lags.

IV. Tanner's Models

Tanner (1975) examined various predator-prey data sets from the literature and measured the intrinsic rate of growth (r) for both species. Tanner hypothesized that under certain conditions the relative growth rates of the predator and prey will determine stability of the interactions.

Tanner proposed 4 models (A, B, C, D) to explain predator-prey interactions, with each model having slightly different assumptions.

Tanner's Model A

• Prey population growth:

where,

w =	maximum predation rate.
D =	constant determining how fast the functional response increases at low densities of prey (proportional to the time required for the predator to search for and find a prey).

Note: the 2nd part of the equation describes a Type 2 functional response.

• Predator population growth:

where,

s =	intrinsic growth rate for the predator.
J =	the number of herbivores required to support 1 predator at equilibrium (i.e., when P = H/J)

Note: describes logistic growth in which K is set by the # of prey or H/J.

Possible outcomes of Tanner's Model A:

1. Stable node (equilibrium point) or stable focus (dampened oscillations).

2. Stable limit cycles (coupled oscillations) - versus Neutral Limit Cycle of L-V model.

4. Unstable cycle, which leads to extinction of prey and predator.

Assumptions of Tanner's model A

1. The two populations inhabit the same area so densities are proportional to numbers;

2. There is no time lag in responses of either population to changes;

3. The prey's food supply is never so overexploited as to reduce food production;

4. The prey is the sole food of the predator;

5. The mortality rate of the predator from causes other than starvation is constant.

Properties affecting stability in Tanner's models:

1. Presence and strength of self-limitation on prey growth.

2. Relative length of predator's searching time, which affects the steepness with which the predation rate increases towards its maximum.

3. Presence or absence of self-limitation on predator's growth (other than their food).

4. The presence or absence of intraspecific competition for food among the predators.

5. The ratio of the predator's intrinsic rate of growth to that of the prey.

6. The presence of one or two habitats.

Note: in the absence of competition for food by the predators, the ratio of growth rates has no effect on stability; this interplay between competition and growth rates is not transparent nor are the results obvious on biological grounds (Tanner 1975).

V. In-class Exercises

VI. Problem Set

VII. Selected References:

• Tanner, J. T. 1975. The stability and the intrinsic growth rates of prey and predator populations. Ecology 56:855-867.

Revised: 19 August 2002