Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction
Pavel A. Shamanaev
Olga S. Yazovtseva
Introduction. The article considers the problem of stability of the mathematical model of the trivial equilibrium. The model describes the biocenosis dynamics with the predatorprey type interspecific interaction, which is a nonlinear system of ordinary differential equations with perturbations in the form of vector polynoms. The examined system is considered provided that the birth rate of biological species does not exceed mortality rate.
Materials and Methods. The article states the sufficient conditions for asymptomatic stability. The proof is based on the construction of an operator equation in a Banach space, which connects the solution of the nonlinear system and its linear approximation. The existence of the operator equation solution is proved through using the Schauder fixed point principle. It is shown that there is Brauer local asymptotic equivalence between the solutions of the investigated system and its linear approximation and the differences between the components of the solutions of the nonlinear system and its linear approximation tends to zero evenly with respect to the initial values.
Results. As a case in point, the authors consider the model of the predator-prey type in the case when two species feed on the third one. The conditions for stability and asymptotic stability for a part of the variables of the trivial equilibrium of the abundance dynamics of two predator populations and one prey population under different fertility rates of biological species are given. The graphs of a number of populations with different vaues of the difference between the birth rate and the mortality rate of partucular species are constructed.
Conclusions. Depending on the difference between fertility and mortality of biological species, the population dynamics of two populations of “predators” and one population of “preys” is analyzed over time.
Keywords: predator-prey model, stability in the respect to a part of variables, nonlinear ordinary differential equations, local asymptotic equivalence by Brauer, Schauder fixed point principle
For citation: Shamanaev P. A., Yazovtseva O. S. Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(3):321–332. DOI: https://doi.org/10.15507/0236-2910.028.201803.321-332
Authors’ contribution: П. P. A. Shamanaev – formulating the problem, prooving the theorem; O. S. Yazovtseva – reviewing the relevant literature, presenting and analysis of the research results.
All authors have read and approved the final version of the paper
Received 14.05.2018; revised 29.06.2018; published online 20.09.2018
1. Yazovtseva O. S. [Local component-wise asymptotic equivalence and its application to investigate stability with respect to a part of variables]. Ogarev-online. 2017; 13. (In Russ.)
2. Shamanaev P. A., Yazovtseva O. S. [The sufficient conditions of local asymptotic equivalence of nonlinear systems of ordinary differential equations and its application for investigation of stability respect to part of variables]. Zhurnal Srednevolzhskogo matematicheskogo obshchestva = Journal of Middle Volga Mathematical Society. 2017; 19(1):102–115.
3. Verhulst P. F. Notice sur la loi que la population suit dans son accorois-sement. Corr. Math. et Phys. 1838; 10:113–121.
4. Pearl R. The growth of populations. Quart. Rev. Biol. 1927; 2:532–548.
5. Volterra V. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. Accad. Naz. Lincei. (Ser. 6). 1926; 2:31–113.
6. Bazykin A. D., Berezovskaya F. S., Buriev T. I. [Dynamics of the predator-prey system taking into account saturation and competition]. Faktory raznoobraziya v matematicheskoy ekologii i populyatsionnoy genetike: sb. nauch. tr. = Factors of
7. Manna D., Maiti A., Samanta G. P. Analysis of a predator-prey model for exploited fish populations with schooling behavior. Applied Mathematics and Computation. 2018; 317:35–48.
8. Panday P., Pal N. Samanta S., Chattopadhyay J. Stability and bifurcation analysis of a three-species food chain model with fear. International Journal of Bifurcation and Chaos. 2018; 28(1).
9. Ma Z., Wang S., Wang T., Tang H. Stability analysis of prey-predator system with holling type functional response and prey refuge. Advances in Difference Equations. 2017; 1.
10. Capone F., Carfora M. F., De Luca R., Torcicollo I. On the dynamics of an intraguild predator- prey model. Mathematics and Computers in Simulation. 2018.
11. Yunfeng J. Analysis on dynamics of a population model with predator-prey-dependent functional response. Applied Mathematics Letters. 2018; 80:64–70.
12. Fergola P., Tenneriello C. Lotka-Volterra models: partial stability and partial ultimate boundedness. Biomath. and Related Comput. Prob. Proc. Workshop. 1988. P. 283−294.
13. Ignatev A. O. On global asymptotic stability of the equilibrium of “predator–prey” system in varying environment. Russian Mathematics. 2017; 61(4):5–10.
This work is licensed under a Creative Commons Attribution 4.0 License.