How do generalist consumers coexist over evolutionary time? An explanation with nutrition and tradeoffs

Abstract

Generalist consumers commonly coexist in many ecosystems. Yet, eco-evolutionary theory poses a problem with this observation: generalist consumers (usually) cannot coexist stably. To provide a solution to this theory-observation dissonance, we analyzed a simple eco-evolutionary consumer resource model. We modeled consumption of two nutritionally interactive resources by species which evolve their resource encounter rates subject to a tradeoff. As shown previously, consumers can ecologically coexist through tradeoffs in resource encounter rates; however, this coexistence is evolutionary unstable. Here, we find that nutritional interactions between resources and the shape of acquisition tradeoffs produce very similar evolutionary outcomes in isolation. Specifically, they produce evolutionarily stable communities composed either of two specialists (concave acquisition tradeoff or antagonistic nutrition) or a single generalist (convex acquisition tradeoff or complementary nutrition). Thus, the generalist-coexistence problem remains. However, the combination of nonlinear resource acquisition tradeoffs with nonlinear resource nutritional relationships creates selection forces that can push and pull against each other. Ultimately, this push-pull dynamic can stabilize the coexistence of two competing generalist consumers—but only when we coupled a convex acquisition tradeoff with antagonistic nutrition. Thus, our model here offers some resolution to the generalist-coexistence problem in eco-evolutionary, consumer-resource theory.

Appendix: Evolutionary stability under different resource nutritional interactions and different resource acquisition tradeoffs

Different resource nutritional interactions

Mathematically, evolutionary equilibrium points occur where the derivative of the fitness function with respect to v, evaluated at v = u, is zero. The derivative of the fitness function is given by:

$$ \frac{\partial G}{\partial v}=b{R}_1^{\ast}\left(\mathbf{u}\right)-b{R}_2^{\ast}\left(\mathbf{u}\right)+b\beta {R}_1^{\ast}\left(\mathbf{u}\right){R}_2^{\ast}\left(\mathbf{u}\right)\left[1-2v\right] $$

One can see that the perfect generalist (v = u = 0.5) evaluates to zero and thus is an equilibrium point. Note that the perfect generalist will reduce both resources to the same equilibrium level (assuming resource supply rates are equal).

To evaluate specialist boundaries as equilibrium points, the fitness gradient needs not be 0. Instead, selection can keep a specialist stable if the landscape slopes up toward the specialist. That is, if the landscape has a positive or negative slope when evaluated at u = 1 (resource 1 specialist) or u = 0 (resource 2 specialist), respectively. We find that a specialist strategy is only selected for when resources are strongly antagonistic. When a single species is a specialist, the R ^* of the resource they specialize on will be m/b (remembering that resource encounter rate is 1 for a specialist). The other resource will be at its supply point (S), since it is not consumed. Plugging in the resource equilibrium values and the value of the evolutionary strategy, and simplifying, shows that a specialist becomes an evolutionary equilibrium point whenever β < 1/S − b/m. Numerical simulations show that this is the same condition for the existence of the two additional interior equilibrium points. Thus, β = 1/S − b/m is a bifurcation point where the stability properties of the model change from a single equilibrium point to three interior equilibrium points.

The curvature of the landscape at an equilibrium point allows us to determine whether the point is invadable or not (ESS). It is evaluated using the second derivative of the G-function with respect to the evolutionary variable v. This is given by

$$ \frac{\partial^2G}{\partial {v}^2}=-2b\beta {R}_1^{\ast}\left(\mathbf{u}\right){R}_2^{\ast}\left(\mathbf{u}\right) $$

Since R ^*s are always positive, the landscape is concave down (evolutionary maximum) when β > 0 (complementary resources), concave up (evolutionary minimum) when β < 0 (antagonistic resources), and has no curvature (evolutionary neutrality) when β = 0 (substitutable resources).

Whenever β < 0, two specialists are always evolutionarily stable. Substituting R ₁ ^* = R ₂ ^* = m/b and evaluating the derivative of the fitness function gives a negative value (b β (m/b)²) at u = 0 and a positive value (−b β (m/b)²) at u = 1. Recall that β is negative when resources are antagonistic.

Different resource acquisition tradeoffs

In this scenario, the fitness gradient is given by:

$$ \frac{\partial G}{\partial v}=b{R_1}^{\ast}\left(\mathbf{u}\right)-b{R_2}^{\ast}\left(\mathbf{u}\right){\left(1-{v}^{\alpha}\right)}^{\left(1/\alpha -1\right)}{v}^{\alpha -1} $$

The perfect generalist is located at u = (1/2)^1/α. At this point, the fitness gradient evaluates to 0, demonstrating that the perfect generalist is an evolutionary equilibrium.

To evaluate the stability of the specialist strategies, we can calculate the derivate at the boundary values (u = v = 0 and u = v = 1). However, since this expression is undefined at these points, we take the limit of this derivative as these values are approached. Evaluating the limit as v = u approaches 1, and the associated R ₁ ^* = m/b and R ₂* = S, gives m for α < 1 (selection for specialist), m − bS for α = 1 (selection depends on which resource is more abundant), and –∞ for α > 1 (selection for generalist). Likewise, evaluating the limit of the derivative as u = v approaches 0, and the associated R ₁ ^* = S and R ₂ ^* = m/b, gives –∞ for α < 1 (selection for the specialist), bS − m for α = 1 (selection depends on which resource is more abundant), and bS for α > 1 (selection for generalist).

The second derivative is given by

$$ \frac{\partial^2G}{\partial {v}^2}=b{R}_2^{*}\left(\mathbf{u}\right)\left[\left(\frac{1}{\alpha }-1\right)\alpha {\left(1-{v}^{\alpha}\right)}^{\left(1/\alpha -2\right)}{v}^{2\left(\alpha -1\right)}-\left(\alpha -1\right){v}^{\left(\alpha -2\right)}{\left(1-{v}^{\alpha}\right)}^{\left(1/\alpha -1\right)}\right] $$

All terms in this equation are always positive other than the (1/α − 1) and (α − 1) terms. One can see that if α > 1 (convex tradeoff), the curvature of the adaptive landscape will be negative (evolutionary maximum), if α = 1 (linear tradeoff), it will be zero (neutral evolutionary point), and if α < 1 (concave tradeoff), it will be positive (evolutionary minimum).

To show that two specialists are stable when the tradeoff is concave (α < 1), we can evaluate the limit of the fitness gradient with two resident specialist strategies, as each specialist point is approached. The right hand side (resource 1 specialist) evaluates to m. The left hand side (resource 2 specialist) evaluates to –∞.