## Introduction

The influence of dispersal on population dynamics is a field of intense current interest to both empirical and theoretical ecologists (Kareiva & Wennergren 1995). One particularly intensely studied group of models are collectively termed coupled map lattices (CMLs). Here, a group of distinct populations are each represented by simple equations which convert the current local population density to that at the next generation. Conceptually, these populations are placed on the nodes of a (normally square) lattice. The equations are linked by transport terms which represent dispersal of individuals between the local populations in the system (Allen 1975; Hassell, Comins & May 1991; Hastings 1992; Rohani, May & Hassell 1995; Ruxton & Rohani 1996; Bascompte & Sole 1998). Despite the proliferation of spatial models in recent years, dispersal is generally modelled in a very simple way. In the overwhelming majority of cases, dispersal is assumed to be density-independent, with a constant fraction of each population dispersing at each generation (but see McCauley, Wilson & de Roos 1993; Scheuring & Janosi 1996 and Ruxton 1996a for exceptions). From an evolutionary viewpoint, we would predict, however, that migration should be density dependent, with a higher fraction of individuals leaving a particular site when the local population density is high. Furthermore, we will argue below that dispersal should also be dependent on other factors at the current site which affect fitness. Our overall objective is to explore the effect of this ‘fitness-driven dispersal’ on population dynamics.

In common with several previous studies, let us consider the case where each generation can be split into two distinct phases: demographic processes and dispersal events. This temporal segregation is biologically plausible for species such as butterflies where most of the lifecycle is spent in relatively immobile immature states, followed by a final highly mobile egg-laying stage. Consider a newly-emerging butterfly which is faced with the choice of remaining in its current habitat or dispersing to another before laying its eggs. To maximize the fitness of its genes, it should remain in its current site if its expectation is that it could obtain more viable offspring there than if it dispersed to another habitat. Say there are *M* female butterflies in the current site, and each lays λ eggs which hatch into females. Competition for resources is likely to introduce density-dependence in juvenile survival. We follow convention and express this by saying that the probability of each of these offspring reaching maturity is a decreasing function of *M* (the number of adults which laid eggs at this site). For example, let this juvenile survival probability *P*(*M*) be given by

where the parameter *a* scales the carrying capacity, and *b* represents the strength and form of competition (the limit *b* → 1 is considered to represent contest competition and *b* → ∞ to represent scramble; see Doebeli 1995 for a fuller discussion). Now, if the butterfly stays in the current site, then its expected fitness return will be λ*P*(*M*). The butterfly should only disperse if it can increase its expected fitness return by doing so. For all but the most trivially simple of systems, this expectation cannot be calculated analytically. However, clearly it will depend on the probability of the individual reaching another population, the number of other adults laying eggs in that population and other aspects of the environment at that location (represented by the values of λ, *a and b*). In nature, individuals will never have sufficiently accurate information to make a precise estimate of conditions on other sites; hence, dispersal may be well represented by a rule of thumb using only information about local conditions. One reasonable rule to use would be for individuals to disperse if their expected fitness return on the current patch falls below some fraction (Δ) of the maximum (λ): i.e. individuals only disperse if P(*M*) < Δ. In population terms, this means that there is a critical population density (*M*_{c}) which satisfies

If the current population density is lower than this, then there will be no emigration. However, if the population density is higher than *M*_{c}, then individuals will leave until the population density falls below the threshold. Thus, from this fitness argument, dispersal will be density-dependent. We can also see that it is related to the constants *a and b* which represent the environmental conditions on the site. This conclusion stands in contrast to the assumption in previous models of density-dependent dispersal, where the density-dependence in the dispersal function was unrelated to demographic parameters (even though these factors control the effect of competitor density on fitness). For example, the model of Scheuring & Janosi (1996) has a threshold parameter *k* (directly analogous to our *M*_{c}) which they vary whilst keeping the carrying capacity constant. From our argument above, this is equivalent to varying individuals’ criterion for moving (represented by Δ above). However, varying *k* in a linear fashion causes a strongly non-linear change in Δ, so Scheuring & Janosi's results are difficult to interpret in terms of the effect of changing movement criteria by individuals on population dynamics.

The aim of this paper will be to use a simple mathematical model of a collection of linked populations, to explore how changes in the propensity of individuals to disperse is related to the local density and how these changes affect population dynamics. Unlike any previous models, the propensity of individuals to disperse will also be related (through the fitness argument above), not only to the local population density, but also to other aspects of the quality of the current site (such as its carrying capacity) which relate the local population density to reproductive success. Furthermore, the site which individuals finally settle on will also depend on the same fitness considerations.