## Predator Aggregation

Another mechanism that promotes prey-predator coexistence in spatially heterogeneous environments is the tendency of predators to aggregate in patches where prey abundance is high. If u denotes the proportion of predators in patch i, the Lotka-Volterra model for two patches is then described by the following set of equations dRi

— = e, A,R, u, C + ej AjRjUjC - mC dt i i i i 2 2 2 2

provided the predator mortality rate is patch independent. Here Ri is prey abundance in patch i, and all other parameters have the same meaning as those in the Lotka-Volterra model, but they are patch dependent now. The above model assumes that prey do not move between the two patches. When predator distribution (u) is fixed, the above model has no interior equilibrium and the prey with smaller value of ri/(uiAi) is always driven to extinction. This is documented in Figure 10a where the second prey type (dashed line) is outcompeted by the first prey (solid line). This indirect interaction is

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Figure 10 The effect of predator aggregation on population dynamics [13]. (a) Shows the extinction of the competitively weaker prey species due to apparent competition when predator preferences for the two prey are fixed (u1 = u2 = 0.5). (b) Shows the coexistence of all species when predator preferences are adaptive. Parameters: r1 = 1.5, r2 = 0.5, m = 0.2, e1 =0.15, e2 = 0.1, A1 = A2 = 1.

mediated by predators (dot line). This kind of indirect competition between the two prey populations, which is mediated by the shared predators, is called apparent competition.

The above analysis assumes that the proportion of predators u in each patch is fixed and independent of prey abundances. Now let us consider the situation where predator patch preferences are adaptive. If predator fitness is measured by the per capita predator population growth rate dC/(Cdt) then predators will aggregate in patch 1 (ui = 1 and u2 = 0) if e1X1R1> e2X2R2 and in patch 2 if the opposite inequality holds (here we neglect the travel time between the two patches). This makes predator preferences for either patch dependent on prey abundances and predators switch between the two patches. Predator switching then changes population dynamics of model [13]. It can be proved that prey dynamics get synchronized and prey-predator population dynamics in both patches are described by the Lotka-Volterra-like cycles (Figure 10b). In particular, both prey populations coexist with predators indefinitely. This clearly shows that predator aggregation can promote species coexistence without necessarily leading to an equilibrium. In this example, adaptive predator switching relaxes apparent competition between the two prey because at low prey density in one patch predators switch to the other patch. In fact, this type of predator behavior drives the two prey populations to the levels where predator fitness is the same in both patches and predators will distribute across both patches following the 'ideal free distribution'.

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