J. M. Cushing

Integrodifferential equations and delay models in population dynamics

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 Delay Logistic Equations.- 3.2 The Logistic Equation with a Constant Time Lag.- 3. 3 Some Other Models.- 3.4 Some General Results.- 3.5 A General Instability Result.- 3.6 The Stabilizing Effect of Delays.- 4: Stability and Multi-Species Interactions with Delays.- 4.1 Volterras Predator-Prey Model with Delays.- 4. 2 Predator-Prey Models with Density Terms.- 4.3 Predator-Prey Models with Response Delays to Resource Limitation.- 4.4 Stability and Vegetation-Herbivore-Carnivore Systems.- 4.5 Some Other Delay Predator-Prey Models.- 4.6 The Stabilization of Predator-Prey Interactions.- 4.7 A General Predator-Prey Model.- 4.8 Competition and Mutualism.- 4.9 Stability and Instability of n-Species Models.- 4.10 Delays Can Stabilize an Otherwise Unstable Equilibrium.- 5: Oscillations and Single Species Models with Delays.- 5.1 Single Species Models and Large Delays.- 5.2 Bifurcation of Periodic Solutions of the Delay Logistic.- 5.3 Other Results on Nonconstant Periodic Solutions.- 5.4 Periodically Fluctuating Environments.- 6: Oscillations and Multi-Species Interactions with Delays.- 6.1 A General Bifurcation Theoren.- 6.2 Periodic Oscillations Due to Delays in Predator-Prey Interactions..- 6.3 Numerically Integrated Examples of Predator-Prey Models with Delays.- 6.4 Oscillations and Predator-Prey Models with Delays.- 6.5 Two Species Competition Models with Linear Response Functionals.- 6.6 Two Species Mutualism Models with Linear Response Functionals.- 6.7 Delays in Systems with More than Two Interacting Species.- 6.8 Periodically Fluctuating Environments.- 7: Some Miscellaneous Topics.- References.

Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model.

We perform sensitivity analyses on a mathematical model of malaria transmission to determine the relative importance of model parameters to disease transmission and prevalence. We compile two sets of baseline parameter values: one for areas of high transmission and one for low transmission. We compute sensitivity indices of the reproductive number (which measures initial disease transmission) and the endemic equilibrium point (which measures disease prevalence) to the parameters at the baseline values. We find that in areas of low transmission, the reproductive number and the equilibrium proportion of infectious humans are most sensitive to the mosquito biting rate. In areas of high transmission, the reproductive number is again most sensitive to the mosquito biting rate, but the equilibrium proportion of infectious humans is most sensitive to the human recovery rate. This suggests strategies that target the mosquito biting rate (such as the use of insecticide-treated bed nets and indoor residual spraying) and those that target the human recovery rate (such as the prompt diagnosis and treatment of infectious individuals) can be successful in controlling malaria.

Bifurcation analysis of a mathematical model for malaria transmission

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 an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow a logistic population model, with humans having immigration and disease‐induced death. We define a reproductive number,

Periodic Time-Dependent Predator-Prey Systems

R_0

NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS'

, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease‐free equilibrium is locally asymptotically stable when

Some Discrete Competition Models and the Competitive Exclusion Principle

R_0 1

Two Species Competition in a Periodic Environment

. We prove the existence of at least one endemic equilibrium point for all

Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles

R_0 > 1

A Periodically Forced Beverton-Holt Equation

. In the absence of disease‐induced death, we prove that the tran...

A predator prey model with age structure

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