Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T12:15:18.646Z Has data issue: false hasContentIssue false
  • English
  • Français

Predator-Prey Interactions, Age Structures and DelayEquations

Published online by Cambridge University Press:  07 February 2014

M. Mohr ,
M. V. Barbarossa  and
C. Kuttler
M. Mohr*
Affiliation:
University of Heidelberg, Institute of Applied Mathematics, D-69120 Heidelberg, Germany
M. V. Barbarossa
Affiliation:
Bolyai Institute, University of Szeged, H-6720 Szeged, Hungary
C. Kuttler
Affiliation:
Institute of Mathematics, Technische Universität München, D-85748 Garching, Germany
*
⋆ Corresponding author. E-mail:marcel.mohr@bioquant.uni-heidelberg.de
Get access

Abstract

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

Keywords

predator-preyage structurepopulation dynamicsdelay differential equationsneutral equationsRosenzweig-MacArthur model
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 1: Issue dedicated to Michael Mackey , 2014 , pp. 92 - 107
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abrams, P. A., Ginzburg, L. R.. The nature of predation: prey dependent, ratio dependent or neither? Trends Ecol. Evol., 15(8) (2000), 337–341. CrossRefGoogle ScholarPubMed
Bartlett, M. S.. On theoretical models for competitive and predatory biological systems. Biometrika, 44(1-2) (1957), 27–42. CrossRefGoogle Scholar
Beretta, E., Kuang, Y.. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal., 33(5) (2002), 1144–1165. CrossRefGoogle Scholar
Bocharov, G. A., Hadeler, K. P.. Structured population models, conservation laws, and delay equations. J. Diff. Equa., 168(1) (2000), 212–237. CrossRefGoogle Scholar
Castellazzo, A., Mauro, A., Volpe, C., Venturino, E.. Do demographic and disease structures affect the recurrence of epidemics? Math. Model. Nat. Phenom., 7(3) (2012), 28–39. CrossRefGoogle Scholar
Crouse, D. T., Crowder, L. B., and Caswell, H.. A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology, 68(5) (1987), 1412–1423. CrossRefGoogle Scholar
Cushing, J. M., Saleem, M.. A predator prey model with age structure. J. Math. Biol., 14(2) (1982), 231–250. CrossRefGoogle Scholar
D’Onofrio, A.. Tumor-immune system interaction: modeling the tumor-stimulated proliferation of effectors and immunotherapy. Math. Mod. Meth. Appl. Sci. 16(8) (2006), 1375–1401. CrossRefGoogle Scholar
L. Edelstein-Keshet. Mathematical models in biology. SIAM, New York, 1988.
L. C. Evans. Partial differential equations. AMS, Providence, 1998.
Frasson, M. V. S.. On the dominance of roots of characteristic equations for neutral functional differential equations. Appl. Math. Comput., 214(1) (2009), 66–72. Google Scholar
Gopalsamy, K., Zhang, B. G.. On a neutral delay logistic equation. Dynam. Stabil. Syst., 2 (1988), 183–195. CrossRefGoogle Scholar
Gourley, S. A. and Kuang, Y.. A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol., 49(2) (2004), 188–200. CrossRefGoogle ScholarPubMed
Gurtin, M. E., MacCamy, R. C.. Non-linear age-dependent population dynamics. Arch. Ration. Mech. An., 54(3) (1974), 281–300. CrossRefGoogle Scholar
Gurtin, M. E., MacCamy, R. C.. Some simple models for nonlinear age-dependent population dynamics. Math. Biosci., 43(3) (1979), 199–211. CrossRefGoogle Scholar
Hastings, A.. Age-dependent predation is not a simple process. I. continuous time models. Theor. Popul. Biol., 23(3) (1983), 347–362. CrossRefGoogle Scholar
Hastings, A.. Delays in recruitment at different trophic levels: Effects on stability. J. Math. Biol., 21 (1984), 35–44. CrossRefGoogle ScholarPubMed
Hbid, M. L., Sanchez, E., Bravo De La Parra, R.. State-dependent delays associated to threshold phenomena in structured population dynamics. Math. Mod. Meth. Appl. Sci., 17(6) (2007), 877–900. CrossRefGoogle Scholar
Kacha, A., Hbid, M. H., Bravo de la Parra, R.. Mathematical study of a bacteria-fish model with level of infection structure. Nonlinear Anal. Real, 10 (2009), 1662–1678. CrossRefGoogle Scholar
Kuang, Y.. On neutral delay logistic Gause-type predator-prey systems. Dynam. Stabil. Syst., 6(2) (1991), 173–189. CrossRefGoogle Scholar
A. J. Lotka. Elements of physical biology. Williams & Wilkins, Princeton, N. J., 1925.
R. May. Complexity and stability in model ecosystems. Princeton University Press, 1973.
M. Mohr. On predator-prey models with delay due to maturation. Master’s thesis, Technische Universität München, Munich, 2012.
Nisbet, R. M., Gurney, W. S. C.. The systematic formulation of population models for insects with dynamically varying instar duration. Theor. Popul. Biol., 23(1) (1983) 114–135. CrossRefGoogle Scholar
Novoseltsev, V. N., Novoseltseva, J. A., Yashin, A. I.. What does a fly’s individual fecundity pattern look like? The dynamics of resource allocation in reproduction and ageing. Mech. Ageing Dev., 124 (5) (2003) 605–617. CrossRefGoogle ScholarPubMed
Nunney, L.. The effect of long time delays in predator-prey systems. Theor. Popul. Biol., 27 (1985), 202–221. CrossRefGoogle ScholarPubMed
Rosenzweig, M. L., MacArthur, R. H.. Graphical representation and stability conditions of predator-prey interactions. Am. Nat., 97 (895) (1963) 209–223. CrossRefGoogle Scholar
Ross, G. G.. A difference-differential model in population dynamics. J. Theor. Biol., 37(3) (1972), 477–492. CrossRefGoogle Scholar
Sharpe, F. R., Lotka, A. J.. A problem in age distribution. Philos. Mag. Ser. 6, 21 (1911), 435–438. CrossRefGoogle Scholar
H. Smith. An introduction to delay differential equations with applications to the life sciences. Springer, New York, 2011.
Solomon, M. E.. The natural control of animal populations. J. Anim. Ecol., 18 (1949), 1–35. CrossRefGoogle Scholar
Venturino, E.. Epidemics in predator–prey models: disease in the predators. Math. Med. Biol., 19(3) (2002), 185–205. CrossRefGoogle ScholarPubMed
Volterra, V.. Variazioni e fluttuazioni del numero d’individui in specie conviventi. Mem. Accad. Lincei Roma, 2 (1926), 31–113. Google Scholar
Wang, W., Chen, L.. A predator-prey system with stage-structure for predator. Comput. Math. Appl. 33(8) (1997), 83–91. CrossRefGoogle Scholar
Williams, K. S., Simon, C.. The ecology, behavior, and evolution of periodical cicadas. Annu. Rev. Entomol., 40(1) (1995), 269–295. CrossRefGoogle Scholar