Coexistence and displacement in consumer-resource systems with local and shared resources

Abstract

Competition for local and shared resources is widespread. For example, colonial waterbirds consume local prey in the immediate vicinity of their colony, as well as shared prey across multiple colonies. However, there is little understanding of conditions facilitating coexistence vs. displacement in such systems. Extending traditional models based on type I and type II functional responses, we simulate consumer-resource systems in which resources are “substitutable,” “essential,” or “complementary.” It is shown that when resources are complementary or essential, a small increase in carrying capacity or decrease in handling time of a local resource may displace a spatially separate consumer species, even when the effect on shared resources is small. This work underscores the importance of determining both the nature of resource competition (substitutable, essential, or complementary) and appropriate scale-dependencies when studying metacommunities. We discuss model applicability to complex systems, e.g., urban wildlife that consume natural and anthropogenic resources which may displace rural competitors by depleting shared prey.

Appendices

Appendix 1: Complementary resources and partially selective diet

This appendix describes the derivation of rules governing consumption of complementary resources when consumers use an optimal foraging strategy and have a type II functional response. Table 1 of the main text shows equations for isolegs, lines in the resource state space where opportunistic and optimal foraging strategies would yield equal gain, which are based on these rules. Figure 6 provides a visual of the isolegs in the resource state space, and the shape of zero net growth isoclines (ZNGIs) for optimal vs. opportunistic foraging.

Fig. 6

Zero net growth isoclines (ZNGIs) and isolegs in a type II system with complementary resources. The shape of the ZNGI depends on the foraging strategy employed by consumers: optimal foraging (dotted red line) vs. opportunistic foraging (solid red line). Isolegs divide the resource state space into three regions in which the optimal forager will have different probabilities of accepting resources. The probability of accepting an item of the local resource is p ₁ and the probability of accepting an item of the shared resource is p ₂

Under a given level of consumption (C _i , where i ∈ {1, 2}), the fitness value of consuming more of the local resource, R ₁, or shared resource, R ₂, is given by

$$ \frac{\frac{dN^j}{dt}}{dC_1}=\frac{\alpha {bC}_1^{\alpha }{C}_2^{\beta }}{C_1} $$

(9)

$$ \frac{\frac{dN^j}{dt}}{dC_2}=\frac{\alpha {bC}_1^{\alpha }{C}_2^{\beta }}{C_2} $$

(10)

Equations (9) and (10) are derived from Eq. (3) of the main text. An encountered item of the shared resource should be rejected if greater fitness can be had by searching for and handling an item of the local resource than by simply handling an item of the shared resource. Therefore, consumers foraging optimally will reject an item of the shared resource if

$$ \frac{a_1{R}_1\frac{\frac{dN^j}{dt}}{dC_2}}{1+{a}_1{R}_1{h}_1}>\frac{\frac{\frac{dN^j}{dt}}{dC_1}}{h_2} $$

(11)

Equation (11) can be rewritten as

$$ \frac{\alpha {a}_1{R}_1{C}_2}{1+{a}_1{R}_1{h}_1}>\frac{\beta {C}_1}{h_2} $$

(12)

Thus, the likelihood of being partially selective increases with α, a ₁, R ₁, h ₂, and C ₂. The likelihood of being partially selective decreases with β, h ₁, and C ₁. A similar equation can be derived for when an encountered item of the local resource should be rejected. In the event that the condition in Eq. (12) is satisfied, then the forager should modulate its consumption of the shared resource so that the condition is met with equality. Therefore, C ₂ should be determined by C ₁ according to the following relationship (found by setting the above equation equal and solving for C ₂ as a function of C ₁):

$$ {C}_2=\frac{\beta \left(1+{a}_1{R}_1{h}_1\right)}{\alpha {a}_1{R}_1{h}_2}{C}_1 $$

(13)

Conditions governing consumption in a system with optimal foraging can be visualized as isolegs, lines that divide the resource state space into regions wherein consumers use different foraging strategies. Substituting the consumption rates when feeding is opportunistic, and rearranging Eq. (13) yields the isolegs separating when the forager should be an opportunist, and when it should be partially selective (Table 1). Between the isolegs, consumers forage opportunistically, consuming resources as encountered. At high values of one resource, consumers reject consumption of some of the more abundant resource to increase consumption on the more limiting resource (Fig. 6). We observe that the main differences between optimal and opportunistic foraging in a type II system is that the shape of the ZNGIs is such that more of the limiting resource is required at high values of the non-limiting resource when foraging is opportunistic (Fig. 6).

Appendix 2: Simulation parameters

This appendix explains the rationale for choosing simulation parameters. The average death rate of the two populations (0.7) is based on the estimated death rate of 0.7 for black-crowned night herons (Nycticorax nycticorax) in their first year (Henny 1972). Conversion efficiency rates of turning resources into consumers (0.1 or 0.2) are also based on life history parameters of black-crowned night herons. For Fig. 5 with complementary resources, d ^A = 0.4 and d ^B = 0.9. In this case, more extreme parameter values were used to illustrate scenarios that would result in consumer displacement.

Consumption multiplied by conversion efficiency yields per capita increase in population size. Therefore, maximum birth rate will be conversion efficiency multiplied by the carrying capacity of the resources. For example, for essential resources in Table 2, conversion efficiency (0.1) multiplied by the maximum resource carrying capacity (16) yields per capita birth rate 1.6 (3.2 per pair). Black-crowned night herons have one to five eggs per clutch, with average 2.9 eggs per clutch, and average annual mean hatching success of nests of 56% (Hothem and Hatch 2004). From these life history values, we determine that a maximum birth rate of 3.2 per pair is reasonable.