TY - GEN

T1 - A Darwinian Ricker Equation

AU - Cushing, Jim M.

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - The classic Ricker equation xt + 1= bxtexp (- cxt) 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.

AB - The classic Ricker equation xt + 1= bxtexp (- cxt) 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.

KW - Chaos

KW - Darwinian Ricker equation

KW - Evolutionary game theory

KW - Ricker equation

UR - http://www.scopus.com/inward/record.url?scp=85101534572&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85101534572&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-60107-2_10

DO - 10.1007/978-3-030-60107-2_10

M3 - Conference contribution

AN - SCOPUS:85101534572

SN - 9783030601065

T3 - Springer Proceedings in Mathematics and Statistics

SP - 231

EP - 243

BT - Progress on Difference Equations and Discrete Dynamical Systems - 25th ICDEA, 2019

A2 - Baigent, Steve

A2 - Bohner, Martin

A2 - Elaydi, Saber

PB - Springer

T2 - 25th International Conference on Difference Equations and Applications, ICDEA 2019

Y2 - 24 June 2019 through 28 June 2019

ER -