Conflict between dynamical and evolutionary stability in simple ecosystems

Abstract

Here, we address the essential question of whether, in the context of evolving populations, ecosystems attain properties that enable persistence of the ecosystem itself. We use a simple ecosystem model describing resource, producer, and consumer dynamics to analyze how evolution affects dynamical stability properties of the ecosystem. In particular, we compare resilience of the entire system after allowing the producer and consumer populations to evolve to their evolutionarily stable strategy (ESS) to the maximum attainable resilience. We find a substantial reduction in ecosystem resilience when producers and consumers are allowed to evolve compared to the maximal attainable resilience. This study illustrates the inherent difference and possible conflict between maximizing individual-level fitness and maximizing resilience of entire ecosystems.

Appendix

Stability of ecological equilibrium

We classify the stability of the interior equilibrium. The elements of the Jacobian matrix at equilibrium (8) from the characteristic equation

$$ a_{3} \lambda^3+a_{2} \lambda^2+a_{1} \lambda+a_0=0. $$

(14)

In order to be a stable equilibrium, the coefficients must satisfy the following Routh-Hurwitz criteria (May 1973): a _n > 0 and a ₂ a ₁ > a ₃ a ₀. It is easy to show that a ₃ > 0 and a ₂ > 0. For the other coefficients, a ₁ > 0 if $\frac {I \Omega (k l + b)}{q + \Omega \Psi } > m b$ and a ₀ > 0 if I Ω > m(q + ΩΨ). The second Routh-Hurwitz criterion, a ₂ a ₁ > a ₃ a ₀, can be simplified to kl > 0, which is always true since both of these parameters are always (+).

Interpreting the CSS-example, location of producer CSS with producer evolution only

The CSS s _P of the producer is partly determined by the consumer strategy s _H because the consumer trait s _H determines which s _P strategies get grazed on most heavily. In this model, the producers are limited by grazing, which means that although there is a trade-off between nutrient uptake and susceptibility to grazing, the producer CSS strategy mainly conforms to what strategy reduces grazing the most (while keeping nutrient uptake as high as possible). For example, in Fig. 8, the producer CSS strategies are all located on the tails of the grazing versus s _P curve. Note that no consumers in the system leads to runaway selection for maximum nutrient uptake of the producers.