Multiple Attractors, Saddles, and Population Dynamics in Periodic Habitats

Abstract

1122 S. M. Henson et al.periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of Tribolium (flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable 'saddle cycle' separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.

Population models in almost periodic environments

“…Using these models, we study the effects of almost periodic forcing of the carrying capacity, the demographic characteristic and survival rates of species on the long-term behavior of discretely reproducing populations. Others have studied the effects of periodic forcing of the carrying capacity, the demographic characteristic and survival rates of species on population dynamics [8,9,10], [12,13,14,15,16], [20,21,22,23,24], [28,29,30,31], and [34,35,36,37]. Typically, such periodically forced models support attracting cycles that are either enhance via resonance or diminished via attenuance.…”

“…Despite the fact that all natural populations suffer temporal environmental fluctuations on some scale, experimental and theoretical studies of population responses to external fluctuations remain relatively rare [8,9,10], [12,13,14,15,16,17], [20,21,22,23,24], [28,29,30], [32], and [35,36,36,38,39,40,41]. To study the effects of fluctuating environments on population dynamics, one can deliberately fluctuate environmental parameters in controlled laboratory experiments.…”

“…Jillson performed such an experiment with laboratory cultures of Tribolium [34]. Others have used mathematical models to study the effects of periodic forcing on population dynamics [8,9,10], [12,13,14,15,16,17], [20,21,22,23,24], [29,30], and [35,36,37,38].…”

Periodic dynamical systems in unidirectional metapopulation models

“…Autonomous metapopulation models are known to support multiple attractors where local populations support single attractors [16,30]. As in the autonomous Ricker equation, increasing values of R i in System (6) force complex bifurcations including period doubling bifurcations, chaotic attractors and multiple attractors with complex basins of attraction [16,30].…”

Population models with periodic recruitment functions and survival rates

“…In periodic environments,p opulation sizes are often either enhanced via resonance or diminished via attenuance [1,6,7,[21][22][23][24][25]39].T he logistic differential and difference equations havebeen used to showthat aperiodic environment is always deleterious. That is, the average of the resulting population oscillations in the periodice nvironment is less than the averageofthe carrying capacity in the corresponding constant environment (attenuance) [4,5,7,36].…”