A Darwinian Ricker Equation

The classic Ricker equation x_{t + 1}= bx_texp (- cx_t) has positive equilibria for b> 1 that destabilize when b> e² after which its asymptotic dynamics are oscillatory and complex. We study an evolutionary version of the Ricker equation in which coefficients depend on a phenotypic trait subject to Darwinian evolution. We are interested in the question of whether evolution will select against or will promote complex dynamics. Toward this end, we study the existence and stability of its positive equilibria and focus on equilibrium destabilization as an indicator of the onset of complex dynamics. We find that the answer relies crucially on the speed of evolution and on how the intra-specific competition coefficient c depends on the evolving trait. In the case of a hierarchical dependence, equilibrium destabilization generally occurs after e² when the speed of evolution is sufficiently slow (in which case we say evolution selects against complex dynamics). When evolution proceeds at a faster pace, destabilization can occur before e² (in which case we say evolution promotes complex dynamics) provided the competition coefficient is highly sensitive to changes in the trait v. We also show that destabilization does not always result in a period doubling bifurcation, as in the non-evolutionary Ricker equation, but under certain circumstances can result in a Neimark-Sacker bifurcation.