Empty Site Pair Model

The empty site model is a neat method of formulating and interpreting a pair model that repre- sents the variable population predator-prey interactions. It naturally incorporates a local birth rule for both species (eliminating the need to postulate a version of equation (5.3)) which is a density- dependent rule in the case of the prey population.

The trick is to consider three species - predators (p), prey (v), and empty sites (x) - as a virtual population of a constant size N = (p) + (v) + (x) instead of two species with a variable total pop- ulation. Dynamically, we still consider the same four events (prey birth, prey death, predator birth and predator death) as seen in the variable population pair model. These events occur at the same rates as previously, with the exception of prey birth which is complicated by the density dependence requirement of nding a neighbouring empty site (x) to the parent into which a prey ospring must be placed: Prey birth into an empty site is therefore at a rate proportional to the number of poten- tial parent prey that neighbour it, and the constant of proportionality is bv. This assumption ts

predator-prey systems in which ospring stay in close contact with their parents. The event set in summarised in gure (5.5).

Birth Death Predator Prey Birth Prey Death Predator

V

X

P

Figure 5.5: Schematic diagram for the empty-site predator prey pair model. Indi- viduals are moved between the three classes of predator (P), prey (V) and empty sites (X) according to the four events shown. The total population size is conserved but the physical population (predators plus prey) is free to vary.

In anticipation of the importance of some migration or mixing of the predator and prey populations, additional to that induced by their dynamic interactions, we introduce one further event. Assume both predators and prey drift randomly into neighbouring empty sites at a constant migration rate per v-x or p-x contact. This form of migration, as desired, will only directly eect the density of pair correlations, and not the singles densities.

As for the variable population approach, this introduces one more parameter - the migration rate , in addition to m and from usual the closure assumption (equation (4.14)) - with which to adjust or describe the pair model's implicit spatial environment. We can think of as either a property of the species or a property of the space itself. For simplicity, the same is used for both predators and prey, but this clearly need not be the case.

Combiningthese ve events produces the dynamicalequations. Because we have reverted to the xed population assumption ((v) + (p) + (x) = N, constant), we can, as with the spatial game examples, deduce single population densities (v)(p) and (x) from the pair totals: m(z) = (vz) + (pz) + (xz) for any z. The result is another ve dimensional system to contrast with equations (5.4):

( _vv) = 2bv (vxv) + (vx) ;2d v(vvp) ;2b p(vvp) + 2 (vxv) + (vx);(vvx) ( _vp) = bv(vxp) ;d v (pvp) + (vp) + bp (vvp);(pvp);(vp) ;d p(vp) + 2(vxp);(pvx);(vpx) ( _vx) = bv (vxx);(vxv);(vx) + dv (vvp);(pvx) ;b p(pvx) + dp(vp) (5.6) + (vxx) + (vvx);(vxv);(xvx);2(vx) + (vpx);(vxp) ( _pp) = 2bp (pvp) + (vp) ;2d p(pp) + 2 (pxp) + (px);(ppx) ( _px) = ;b v(vxp) + dv (pvp) + (vp) + bp(pvx) + dp (pp);(px) + (pxx) + (ppx);(pxp);(xpx);2(px) + (pvx);(vxp)

Numerical investigation of equations (5.6) was undertaken with ( > 0) and without ( = 0) migra- tion. Space was described by equation (4.14) and initial parameter values of m = 10 and = 0. The relatively high value of m (in comparison to a square grid approximation) was chosen because of the expected inuence of potentially ubiquitous empty sites and because m should be interpreted as the maximum number of real (i.e. non-empty site) neighbours any individual may possess. The

minimal value for represents minimal clumping within the population in the sense of Chapter 4. In the case of = 0 the behaviour was found to be qualitatively similar to the mean-eld Lotka- Volterra equations (1.11) when modied by a density-dependent prey birth rate,e.g. b

V !b

V(N ;

V ): Trajectories typically either fell into a stable coexistence equilibrium or else predators were eliminated and the prey grew to the maximum possible size N. Increasing bV or decreasing dV,

which increases the number of predators at equilibrium in the non-spatial model did the same in the pair model. Unlike the non-spatial case, however, the prey equilibrium was also decreased instead of remaining unaected. (It is not surprising that the consequences of altering a parameter are more widespread in the more complex pair model). Increasing dP or decreasing bP similarly also

increased the prey, and decreased the predator, equilibrium populations. Extinction of the predators was found with suciently large or small values here respectively. In all cases studied, the results were apparently unaected by a range of dierent parameter sets and generic initial conditions, suggesting that this behaviour is robust and stable.

Increasing upwards from zero always tended to reduce the number of predators and increase the prey in the simulations studied. Suciently large had the eect of eliminating predators in a similar way to changing parameter values (e.g. decreasing b

V) directly. This is consistent with the

interpretation of as a measure of the degree of spatial clumping within the population (Chapter 4) because predators need contact with prey to survive and reproduce, but prey can exist with or

without any contacts. Therefore in a clumped population, predators might quickly exhaust any local food supply (prey) and then experience diculty nding more prey outside of their local cluster. By the same argument, increasing m alone should increase predator numbers by providing more contactable prey, and this was generally found to be the case.

By contrast, increasing the migration rate away from zero appeared to have very little eect on the system's behaviour, either dynamically or on the nal position of the equilibrium. This is not altogether surprising because we expect increased mixing to destroy pair correlations and force the model to behave more like its mean-eld cousin. But since the pair model already displays mean-eld like behaviour, little change is seen. Figure (5.6) shows typical results of varying and away from zero on the approach to and position of the equilibrium for the predator-prey pair model equations (5.6). 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 prey (V) predator (P) increasing phi a 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 prey (V) predator (P) increasing xi b

Figure 5.6: The typical eect of increasing the spatial clumping parameter and the migration rate on the equilibrium behaviour of the predator-prey-empty site pair model. Both gures were integrated numerically at parameter values bP =

1:0bV = 1:0dV = 1:0dP= 2:0m = 10 and N = 1 (which only eects the overall

scaling) from identical initial conditions (a thoroughly mixed population with all pairs equally represented). Only the latter portion of each trajectory, approaching the equilibrium, is shown. a) represents increasing over 0:00:20:40:60:8 and 0:9 for = 0. The predator population gradually decreases until it can no longer be sustained by the prey, and it eventually becomes extinct (between = 0:8 and = 0:9). b) shows increasing over 0110 and 100 for = 0. Dierent levels of migration have only a minimal eect on the system.

Although only a relatively small sample of parameter space could be explored, no exceptions were found to the general scheme outlined above when generic initial conditions were used. The only case discovered to have slightly dierent dynamics resulted from a highly unusual initial condition where

(vp) = (vx) = (px) = 0. Simulations then predicted predator extinction and a prey equilibrium, but where the equilibrium prey density (v) < N i.e. space is not saturated with prey, even in

the absence of predators. This result is best attributed to implicit `inaccessible' empty space (the non-zero (xx)) into which no ospring can be born (zero (xv)) and which therefore must remain empty. Not even an increase in the migration rate can help because migrants too nd the space inaccessible. Interestingly, this spatial segregation eect was only observed for the initial condition (vp) = (vx) = (px) = 0 and not for any other tested alternatives.

5.4 Discussion

In some respects, the IPD and predator-prey pair models described in this chapter have been dis- appointing because their behaviour has been rather similar to that of their non-spatial cousins. However, even negative results require interpretation, and the interpretation may be that space, as represented in these pair models, is not dynamically very important.

In the case of the IPD, the pair model studied did not predict the higher levels of generosity that have been suggested by some other spatial studies. Reasons why this may be the case were dis- cussed. It is important to emphasize that this does not mean that pair analysis has failed, only that this particular implementation was not able to capture the essential requirements for more generous strategies to ourish. There are many other possible formulations of the IPD game rules and spatial structure assumptions from which alternative pair models could be generated, and such models may give dierent predictions. On the other hand, inability to promote generosity across a range of pair models would positively suggest that larger scale spatial structure is an important factor.

The predator-prey systems highlighted diculties associated with migration and mortality events from the point of view of pair model analysis. These diculties are not insurmountable (working models were derived), but the events do raise questions that do not exist in non-spatial analysis because they explicitly involve alterations to the underlying spatial structure. Again, in practice, this means that many alternative pair models, each associated with dierent assumptions, can read- ily be derived. Only the simplest cases were considered here. Both types of event are likely to be important in many other systems and Chapter 7 discusses the problem in more detail.

In summary, all the pair models examined did produce sensible output, which could be analysed at length to establish quantitatively the behavioural dependencies on the range of model parameters. Furthermore, there are more of these dependencies than in non-spatial models because of the extra parameters associated with spatial structure.