We can add another level of complexity to understanding patterns of foraging among predators.

This is the effect of patch use:

Define patch as : a discrete habitat or object where prey is distributed somewhat uniformly and separated from other prey which occur in other patches.

Again, another model by Charnov (1976), developed to describe the movement of predator from one patch to another. The model describes how long a predator should spend foraging in a patch. Model referred to as Optimal Patch Theory or Marginal Value Theorem.

In general: consumer (predator) should spend more time in higher profitable patches. As patch quality changes due to resource depletion (e.g. bees feeding on flower nectar, insectivorous birds) predator should leave to find another patch.

Question: When should the decision to switch to another patch occur?

Optimum residence time will maximize it’s overall energy intake during a bout of foraging. In doing so, it will maximize the average rate of energy intake for the bout as a whole.

Movement between patches assumes energy intake will be zero.

Question: At what point should forager leave patch?

always moving, little time spent feeding. vs never move – deplete resource

therefore, some intermediate is optimal

Let’s consider:

- food is distributed patchily some patches more valuable than others.

- average time spent travelling between patches is is t_t. t_t is the length of time a forager can expect to spend on average after leaving a patch before finding another.

To maximize rate of energy extracted from the patch it will be a function of : cumulative energy extracted(CEE)/ time.

Some predictions generated from the marginal value theorem.

Thus far we have considered population models which show how populations can be limited by resources (intraspecific interactions) and how interspecific competition can affect population size.

Do predators limit prey and vice-versa?

Sometimes clearly not, other times they appear to be tightly linked. In fact, some predator/prey systems show highly correlated oscillation patterns. Are these patterns due to direct interactions, or are they the result of indirect effects? Why are some systems stable and other not.

If predators do control prey populations and vice-versa, what are the conditions required to achieve that and is there any evidence for this?

Start with a simple model, then modify assumptions to see how predictions of the model change.

We’ll look at case studies to see if the model represents reality in nature.

1. Growth of the prey population is not restricted by the environment. Not realistic because most species have low population growth of very low densities and very high densities. For the first case (low) there is difficulty in finding mates, in the second case (high), food and/or space is limiting.
2. The predators’ efficiency at capturing prey is constant. Again, not realistic because most predators are better at capturing prey at certain densities (i.e. functional response changes).
3. Population responses to increasing or decreasing growth in either the predator or prey is reasonably fast. If not, predator can drive prey extinct or vice versa.

The predator-prey L/V model predicts that the predator and prey populations will oscillate in a periodic fashion – amplitude of oscillation dependent on initial size of the two populations. Any disturbance (external) will change the cycle amplitude. This type of system is referred to as being neutrally stable.

For the model:

N₁ = number of individuals in prey population.

N₂ = number of individuals in predator population

For prey:

y ₁ = rate of removal of prey by predator

r₁ = growth rate of prey population

y ₁N₂ = amount of prey removed by predator

To describe the rate of growth of the prey population:

dN₁/dt = (r₁ - _{y 1}N₂) N₁

= (growth rate of prey - removal rate) (pop. of prey)

= r₁ N₁ -y ₁N₂ N₁ where r₁ and y ₁ are constants

For predator:

birth rate of predator depends on density of prey so,

N₁ = prey population

y ₂ = efficiency of predator (in terms of converting

prey into predator offspring)

N₁ y ₂ = rate of increase of predator population

d₂ = rate of death of predator

dN₂/dt = (N_{1y 2} - d₂) N₂

(birth rate – death rate) x population size

= N₂ N_{1y 2} - d₂ N₂

y ₂ - d₂ are constants

Like in the competition model, we can ask what population size will N₁ and N₂ be at equilibrium?

When dN/dt = 0

For prey:

For predator:

Now since r, d, and y ’_s are constant these curves will be straight lines.

Model also assumes:

1. no density-dependent regulation of predator or prey populations.
2. environment is homogeneous.
3. each prey has equal probability of being eaten.
4. prey density does not affect the probability of prey item being eaten, i.e. type 1 functional response.
5. predator density does not effect probability of predator catching a prey item.