Abstract
In this paper, we study an age-structured model for a consumer population coupled with a system of n ordinary differential equations, which has a constant delay, which describes the evolution of several resources, utilized by the consumer as food. The introduction of the delay in the model makes it distinct from the related traditional consumer-resource models, for example, models for Daphnia population feeding on algae. We determine the steady states of the above mentioned coupled system of equations, and show that basically, there are three types of steady states: the first is the trivial steady state, when the consumer population vanishes and the resources depleted. The second is the case when the consumer population vanishes and some of the resources at their carrying capacity and the remaining, if any, vanish. The third, and last, is the case when both the consumer population and the resources that remain (not depleted), coexist. Our main purpose in this paper is to provide stability results for the aforementioned steady states. We obtain the characteristic equation of the coupled system of equations, and prove that the trivial steady state is unstable, whereas the second type of steady states is locally asymptotically stable only when all of the resources are at the carrying capacity, otherwise, is unstable. The third type of steady states, when at least one resource is depleted, is locally asymptotically stable, under a given condition, otherwise, is unstable. The stability of the third type of steady states, when none of the resources is depleted, will be given in a sequel to this paper.
References
[1] V. Akimenko. Stability analysis of delayed age-structured resource-consumer model of population dynamics with saturated intake rate, Frontiers in Ecology and Evolution, 9, (2021), 1-15.
[2] M. El-Doma. (Ed.) Daphnia: Biology and Mathematics Perspectives, Nova Science Publishers, Inc., New York, 2014. ISBN 978-1-63117-028-7.
[3] L. Abia, O. Angulo, J. Lopez-Marcos, M. Lopez-Marcos. The convergence analysis of a numerical method for a structured consumer-resource model with delay in the resource evolution rate. Mathematics, 8, 1440, (2020), 1-18. Doi: 10.3390/math8091440.
[4] D. Breda, O. Diekmann, S. Maset, R. Vermiglio. A numerical approach for investigating the stability of equilibria for structured population models. Journal of Biological Dynamics, 7, (2013), 4-20. Doi: 10.1080/17513758.2013.789562.
[5] O. Diekmann, M. Gylenberg, J. A. J. Metz, S. Nakaoka, A. M. De Roos. Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example. J. Math. Biol., 61, (2010), 277-318. Doi: 10.1007/s00285-009-0299-y.
[6] O. Diekmann, P. Getto, M. Gylenberg. Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal., 39, (2007) 1023-1069. Doi: 10.1137/060659211.
[7] M. El-Doma. A Size-Structured Population Dynamics Model of Daphnia, Applied Mathematics Letters, 25, (2012), 1041-1044. Doi: 10.1016/j.aml.2012.02.067.
[8] M. El-Doma. Size-Structured Population Models of Daphnia with several algae resources and Unification by characteristic equa-tions, Academia Letters, Article 4094, 2021. https://doi.org/10.20935/AL4094.
[9] Liu, Y., He, Z. Stability results for a size-structured population model with resouces-dependence and inflow. J. Math. Anal. Appl. 360, 665-675, (2009).
[10] J.A.J. Metz, O. Diekmann. (Eds.) The dynamics of physiologically structured populations, Lecture notes in biomathematics, 68, Springer-Verlag, 1986.