The importance of being discrete (and spatial).

*(English)*Zbl 0846.92027In recent years the importance of spatial phenomena has been recognized increasingly in discussions of the ecology and evolution of species in terms of modeling approaches. Each approach has its advantages and disadvantages and leads to unique conclusions. We select three prototypical spatial approaches from among this bestiary and compare the predictions with the null model of no spatial detail. Specifically, our focus is on the comparison of four different approaches to modeling the dynamics of spatially distributed systems:

1. mean field approaches (described by ordinary differential equations) in which every individual is considered to have equal probability of interacting with every other individual;

2. patch models that group discrete individuals into patches without other levels of spatial structure (i.e., individuals interact equally with all individuals in the same patch and there is between-patch migration that treats all other patches equally);

3. reaction-diffusion equations, in which infinitesimal individuals diffuse in space and undergo purely local interactions;

4. interacting particle systems, in which individuals are discrete and space is treated explicitly.

To compare and contrast these four approaches we examined three examples of interactions in spatially distributed populations. The differences we have observed in the four approaches are of interest beyond the models and point to the consequences of particular influences such as localization of interactions or discreteness of individuals.

In the first case where the fitness of one type of individual is enhanced by the presence of the other, all four approaches reach the same conclusion: there is a unique equilibrium that is the limit starting from any initial state in which both species have positive density. More generally, we expect this consensus of opinion to occur in most cases in which the ordinary differential equation has a globally attracting fixed point.

In the second case where two individuals compete for the same resource, the spatial models disagree with the nonspatial ones. The ordinary differential equation has two stable equilibria on the axes and their basins of attraction contain the whole positive quadrant except for a boundary line which is attracted to the interior equilibrium. Similarly, in the patch model, the species that wins depends on the initial densities of the two types. By contrast, in the reaction diffusion equation and in the interacting particle system there is one species that is the winner whenever it is present at a positive density. In the spatial models, the victorious type first establishes itself in some region which then grows linearly in time and covers the entire space. A general picture that covers this case is that when we have two stable equilibria, their relative stability in the reaction-diffusion equation can be determined by examining the speed of the traveling wave connecting the two equilibria. Results of R. Durrett and C. Neuhauser [Ann. Probab. 22, No. 1, 289-333 (1994; Zbl 0799.60093)] and R. Durrett and G. Swindle [Probab. Theory Relat. Fields 98, No. 4, 489-515 (1994; Zbl 0794.60106)] show that the last reasoning is valid for interacting particle systems with fast migration.

1. mean field approaches (described by ordinary differential equations) in which every individual is considered to have equal probability of interacting with every other individual;

2. patch models that group discrete individuals into patches without other levels of spatial structure (i.e., individuals interact equally with all individuals in the same patch and there is between-patch migration that treats all other patches equally);

3. reaction-diffusion equations, in which infinitesimal individuals diffuse in space and undergo purely local interactions;

4. interacting particle systems, in which individuals are discrete and space is treated explicitly.

To compare and contrast these four approaches we examined three examples of interactions in spatially distributed populations. The differences we have observed in the four approaches are of interest beyond the models and point to the consequences of particular influences such as localization of interactions or discreteness of individuals.

In the first case where the fitness of one type of individual is enhanced by the presence of the other, all four approaches reach the same conclusion: there is a unique equilibrium that is the limit starting from any initial state in which both species have positive density. More generally, we expect this consensus of opinion to occur in most cases in which the ordinary differential equation has a globally attracting fixed point.

In the second case where two individuals compete for the same resource, the spatial models disagree with the nonspatial ones. The ordinary differential equation has two stable equilibria on the axes and their basins of attraction contain the whole positive quadrant except for a boundary line which is attracted to the interior equilibrium. Similarly, in the patch model, the species that wins depends on the initial densities of the two types. By contrast, in the reaction diffusion equation and in the interacting particle system there is one species that is the winner whenever it is present at a positive density. In the spatial models, the victorious type first establishes itself in some region which then grows linearly in time and covers the entire space. A general picture that covers this case is that when we have two stable equilibria, their relative stability in the reaction-diffusion equation can be determined by examining the speed of the traveling wave connecting the two equilibria. Results of R. Durrett and C. Neuhauser [Ann. Probab. 22, No. 1, 289-333 (1994; Zbl 0799.60093)] and R. Durrett and G. Swindle [Probab. Theory Relat. Fields 98, No. 4, 489-515 (1994; Zbl 0794.60106)] show that the last reasoning is valid for interacting particle systems with fast migration.

##### MSC:

92D40 | Ecology |

37-XX | Dynamical systems and ergodic theory |

35K57 | Reaction-diffusion equations |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |