The limitation of species range: a consequence of searching along resource gradients

Abstract

Ecological modelers have long puzzled over the spatial distribution of species. The random walk or diffusive approach to dispersal has yielded important results for biology and mathematics, yet it has been inadequate in explaining all phenomenological features. Ranges can terminate non-smoothly absent a complementary shift in the characteristics of the environment. Also unexplained is the absence of a species from nearby areas of adequate, or even abundant, resources. In this paper, I show how local searching behavior-keyed to a density-dependent fitness-can limit the speed and extent of a species' spread. In contrast to standard diffusive processes, pseudo-rational movement facilitates the clustering of populations. It also can be used to estimate the speed of an expanding population range, explain expansion stall, and provides a mechanism by which a population can colonize seemingly removed regions - biogeographic islands in a continental framework. Finally, I discuss the effect of resource degradation and different resource impact/utilization curves on the model.

Fig. 1

A comparison of resource and population densities and fitness measure at steady-state. The resource density, R(x) = 10 − .4(x − 5)², is represented by the diamond or large-block curve, while the population density, u(x), is shown by the solid line. The fitness measure, S = R/(u+h) with h = 1 and C = 4, is given by the dotted line. The measure is uniform over the interval where the population density is non-trivial; elsewhere, it is proportional to the resource.

Fig. 2

Divorcement of a multi-peaked population. This figure demonstrates the impact of either a sudden decimation event or a rising global mortality on a population. In both plots above, the resource curve is signified by the same double-peaked, diamond-marked curve which is piecewise C². As in Fig. 1, the population density and the fitness measure, S = R/(u + h), are illustrated by the solid and dotted lines, respectively. In 2A, h = .8 and C = 2. The population is large enough to occupy a contiguous region. In 2B, C = 3.5. Allopatry has been induced as a result of reduction in population, and the fitness measure exhibits a dip in the center corresponding to the unpopulated zone between the two sub-populations.

Fig. 3

Multiple and unique population distributions. Fig. 3A diagrams a common bifurcation in the possible ways in which a given population can be ideally-distributed under a multi-peaked resource. $U_1^{*}\text{and}{U}_2^{*}$ represent the sub-populations under the first and second peak, respectively. The gray area represents multiple equilibrium distributions for a given population size (along the line $U_1^{*}+U_2^{*}=U_{\mathit{Total}}$). For large populations (corresponding to a common, small contour value C < C_V), there is a unique distribution (the solid line). The dashed segment represents non-contiguous distributions sharing a common contour value. Fig. 3B shows an example of colonial peripatry under fast-time/slow-time dynamics (Section 3.4). The population begins entirely under the first resource peak until it grows to reach the resource valley. All additional growth is directed to the sub-population under peak 2. Once the two populations are contiguous, they both grow together until C = μ/r.

Fig. 4

Expansion of the population. Figures 4A,B give a representative example of how the population distribution changes to occupy the available space when movement, mortality, and growth are on comparable time-scales. The initial distribution is u ≡ 5 on [2, 3.5] and 0 elsewhere (the rectangular aperture). h, R and S as in Figure 1. Fig. 4B presents a cross-sectional view of the distributions in 4A in regular increments of 14.5 time units, shown by the series of expanding, dotted curves. The parabolic solid curve represents the terminal steady-state distribution. During expansion, the leading edge of the distribution has a strong wave-front characteristic with the density rapidly falling away from the current peak or edge as it distribution moves rightward.

Fig. 5

Support of a peripheral population. The resource is identical to the double–peaked resource in Fig. 2 S = R/(u + h). The initial population, u ≡ 5 on [2, 3.5], is limited to the area under the left resource hump. The satellite population is not begun until the parent population achieves a rough optimal distribution that reaches the trough.

Fig. A.1

Compression effect. The population is intitially distributed sinusoidally, u(x, 0) = 4+3 sin(3x), over a parabolic resource, R = 10 −.4(x−5)². the population distribution changes only through movement (r = μ = 0). A.1A: The distribution maintains a wavy pattern for much of its history, with densities near the range boundary particularly resistant to change. A.1B: The fitness landscape is much more responsive, with compression of extrema clearly evident. This same information is shown as a space-time contour plot of fitness in A.1D. A.1C: As an additional measure of compression, the global maximum fitness in Ω monotonically decreases over time.