Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

Abstract We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.

[1]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[2]  Yang Kuang,et al.  Boundedness of Solutions of a Nonlinear Nonautonomous Neutral Delay Equation , 1991 .

[3]  H L Smith,et al.  Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. , 1993, Mathematical biosciences.

[4]  K. Hadeler.  Neutral Delay Equations from and for Population Dynamics , 2007 .

[5]  István Györi,et al.  A neutral equation arising from compartmental systems with pipes , 1991 .

[6]  Y. Kuang.  Qualitative analysis of one- or two-species neutral delay population models , 1992 .

[7]  Yang Kuang,et al.  On neutral delay logistic gause-type predator-prey systems , 1991 .

[8]  Y. Kuang.  On neutral-delay two-species Lotka-Volterra competitive systems , 1991, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[9]  Jun Cao,et al.  Life History Studies of the Flightless Marine Midges Pontomyia spp. (Diptera: Chironomidae) , 1999 .

[10]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[11]  Hal L. Smith.  Equivalent Dynamics for a Structured Population Model and a Related Functional Differential Equation , 1995 .

[12]  Opposite Density Effects of Nymphal and Adult Mortality for Periodical Cicadas , 1984 .

[13]  Kathy S. Williams,et al.  The Ecology, Behavior, and Evolution of Periodical Cicadas , 1995 .

[14]  Jim M Cushing,et al.  STABILITY AND MATURATION PERIODS IN AGE STRUCTURED POPULATIONS , 1981 .

[15]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[16]  Gennady Bocharov,et al.  Structured Population Models, Conservation Laws, and Delay Equations , 2000 .