Heterogeneity in the graphical model

refuges, partial refuges and vertical isoclines

We can start by incorporating into the Lotka-Volterra isoclines some relatively simple types of heterogeneity. Suppose that a portion of the prey population exists in a refuge: for example, shore snails packed into cracks in the cliff-face, away from marauding birds, or plants that maintain a reserve of material underground that cannot be grazed. In such cases, the prey zero isocline rises vertically at low prey densities (again, see Figure 10.11), since prey at low densities, hidden in their refuge, can increase in abundance irrespective of predator density.

Even if predators tend simply to ignore prey in low-density patches, as we have seen in some aggregative responses (see Section 9.6), this comes close to those prey being in a refuge, in the sense that the predators do not (rather than cannot) attack them. The prey may therefore be said to have a 'partial refuge', and this time the prey isocline can be expected to rise almost vertically at low prey abundances.

We saw above, when discussing type 3 functional responses, that such isoclines have a tendency to stabilize interactions. Early analyses of both the Lotka-Volterra and the Nicholson-Bailey systems (and early editions of this textbook) agreed with this conclusion: that spatial heterogeneities, and the responses of predators and prey to them, stabilize predator-prey dynamics, often at low prey densities (Beddington et al., 1978). However, as we shall see next, subsequent developments have shown that the effects of heterogeneity are more complex than was previously supposed: the effects of heterogeneity vary depending on the type of predator, the type of heterogeneity, and so on.

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