1: Introductory Remarks.- 2: Some Preliminary Remarks on Stability.- 2.1 Linearization.- 2.2 Autonomous Linear Systems.- 3: Stability and Delay Models for a Single Species.- 3.1 Logistic Equations.- 3.2 The Equation with Constant Time Lag.- 3. 3 Other Models.- 3.4 General Results.- 3.5 A Instability Result.- 3.6 Stabilizing Effect of Delays.- 4: Multi-Species Interactions 4.1 Volterra's Predator-Prey Model 4. 2 Density Terms.- 4.3 Response Delays to Resource Limitation.- 4.4...

We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by infectious mosquito. They then progress through exposed, infectious, recovered classes, before reentering susceptible class. mosquitoes become bite or humans, once move exposed classes. Both species follow a logistic population model, with having immigration disease‐induced death. define reproductive number,...

A nonlinear demographic model was used to predict the population dynamics of flour beetle Tribolium under laboratory conditions and establish experimental protocol that would reveal chaotic behavior. With adult mortality rate experimentally set high, animal abundance changed from equilibrium quasiperiodic cycles chaos as adult-stage recruitment rates were manipulated. These transitions in corresponded those predicted by mathematical model. Phase-space graphs data together with deterministic...

The general system of differential equations describing predator-prey dynamics is modified by the assumption that coefficients are periodic functions time. By use standard techniques bifurcation theory, as well a recent global result Rabinowitz, it shown this has solution (in place an equilibrium) provided long term time average, predator’s net, unihibited death rate in suitable range. from time-dependent logistic equation for prey (which results absence any predator). Numerical which...

An important component of the mathematical definition chaos is sensitivity to initial conditions. Sensitivity conditions usually measured in a deterministic model by dominant Lyapunov exponent (LE), with indicated positive LE. The measure has been extended stochastic models; however, it possible for (SLE) be when LE underlying negative, and vice versa. This occurs because long‐term average over attractor while SLE stationary probability distribution. property conditions, uniquely associated...

Our approach to testing nonlinear population theory is connect rigorously mathematical models with data by means of statistical methods for time series. We begin deriving a biologically based demographic model. The analysis identifies boundaries in parameter space where stable equilibria bifurcate periodic 2—cycles and aperiodic motion on invariant loops. analysis, stochastic version the model, provides procedures estimation, hypothesis testing, model evaluation. Experiments using flour...

Introduction. Models. Bifurcations. Chaos. Patterns in What We Learned. Bibliography. Appendix.

A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, nonlinear ecological interactions, provided such interactions could correctly identified and modeled. The realization in recent decades result chaos other exotic dynamic behaviors exciting but tantalizing, attributing a particular real to complex dynamics mathematical model proved an elusive goal. We experimentally tested...

A difference equation model, called that Leslie/Gower played a key historical role in laboratory experiments helped establish the "competitive exclusion principle" ecology. We show this model has same dynamic scenarios as famous Lotka/Volterra (differential equation) competition model. It is less well known some anomalous results from seem to contradict principle and dynamics. give an example of non-Lotka/Volterra dynamics are consistent with case.

The net reproductive value n is defined for a general discrete linear population model with non‐negative projection matrix. This number shown to have the biological interpretation of expected offspring per individual over its life time. main result relates population's growth rate (i.e. dominant eigenvalue λ matrix) and shows that stability extinction state (the trivial equilibrium) can be determined by whether less than or greater 1. Examples are given show explicit algebraic formulas often...

1. We experimentally set adult mortality rates, μ a , in laboratory cultures of the flour beetle Tribolium at values predicted by biologically based, nonlinear mathematical model to place regions different asymptotic dynamics. 2. Analyses time-series residuals indicated that stochastic stage-structured described data quite well. Using and maximum-likelihood parameter estimates, stability boundaries bifurcation diagrams were calculated for two genetic strains. 3. The transitions dynamics...

has a unique positive equilibrium K and all solutions with x0 . 0 approach as t !1: This equation (known the Beverton–Holt equation) arises in applications to population dynamics, that context is “carrying capacity” r “inherent growth rate”. A modification of this study populations living periodically (seasonally) fluctuating environment replaces constant carrying capacity by periodic sequence Kt capacities.

1. We identify an unstable equilibrium with a two‐dimensional stable manifold and one‐dimensional in three‐state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers larvae, pupae adults are drawn closely to before approaching asymptotic attractor (a two‐cycle), while other points not. 3. Using two quantitative indices, we examine from Tribolium experiment for evidence predicted node. conclude accounts...

We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in species present) and exclusion one absent). Recent studies have investigated the inclusion of life-cycle stages models as casual mechanism for existence these kinds In paper we investigate role without stages.

Animals and many plants are counted in discrete units. The collection of possible values (state space) population numbers is thus a nonnegative integer lattice. Despite this fact, mathematical models assume continuum system states. complex dynamics, such as chaos, often displayed by continuous-state have stimulated much ecological research; yet discrete-state with bounded size can display only cyclic behavior. Motivated data from experiment, we compared the predictions models. Neither...