Parameter based stability analysis of generalized mathematical model with delay of competition between two species

doi:10.1016/j.amc.2020.125791

Applied Mathematics and Computation

Volume 394, 1 April 2021, 125791

https://doi.org/10.1016/j.amc.2020.125791 Get rights and content

Highlights

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Mathematical model with delay for competition of two species.
•
D-decomposition method.
•
Parameter based stability.
•
Stability regions.

Abstract

Two of the importance of stability analysis is investigating the influence of the parameters on stability of equilibrium points of mathematical model and identifying the region of stability in the plane of the parameters. This study is aimed at identifying region of stability in the plane of parameters and setting conditions of stability of equilibrium points of a generalized mathematical model with delay of two species characterized by an interaction of the type competition. d-decomposition method, direct calculation of eigenvalues and simulation of the model for different numerical values using computing software MATLAB is employed. As a result, different locally asymptotically stability conditions were established and stability regions in the parameter plane were identified for different specific values. The practicality of the method and the results were confirmed by using specific mathematical models of the type under consideration. The parameter based stability analysis conducted on the specific model revealed that the result of this research can be used to find parameter domain for which an interior equilibrium point of a given dynamic system of the type under discussion is locally asymptotically stable.

Introduction

One of the purposes of mathematical models is describing the dynamic processes occurring in the problem of ecology describing interaction among species. Mathematical models developed for describing interaction among species enable computer simulation of the processes under study. Simulation of the dynamic system in turn plays a very important role in the main direction of research in ecology; global ecological process and its structural scheme. Thus organizing experimental results and theoretical studies, with the aim of obtaining an integrated view of the dynamic system can be obtained via mathematical modeling of a system and experimentation of the model by simulation using specific data. An integrated view of a dynamic system is a tool not only for analysis, but for designing management strategies.

Delay differential equations appear in almost all studies involving mathematical models of understanding dynamics of a system. Mathematical models with one delay are usually applied with assumption that the effect of the other delay is very small or insignificant and then can be neglected. However, mathematical models with multiple delays are of great interest in the stability analysis of dynamic systems. Delay dynamic systems with two delays, are common in areas like population interaction, biological systems, neural networks, and SEIR epidemic models [1], [2], [3], [4].

Identifying parameter values that guarantee local stability of equilibrium point(s) of a dynamic system is vital for synthesizing a control system and managing the stability of a system for its proper performance. Constructing the whole region of stability in the plane of two parameters for a certain dynamic system is one of the important areas of research interest especially for investigation of asymptotic stability.

The boundaries of the stability regions can be obtained using several methods including Nyguist criteria [5], Mikhailov Criterion and τ - decomposition methods [6]. But the more general method of constructing stability region is said to be d-decomposition method first developed by a Russian Engineer Neimark [7], [8], [9], [10], [11], [12].

The d-decomposition method is partitioning the parameter plane into stable and unstable regions. To identify the region of stability in the parameter plane, the d-decomposition curves need to be hatched on one side depending on the sign of the principal determinate.

In general, in the course of stability analysis it is essential to investigate the influence of parameters on stability of equilibrium points of a dynamic system and identify a domain of stability in the plane of the parameters. The d-decomposition method is one such mathematical instrument that helps in identifying region of stability in parameter space. Hence, the objective of this study is to investigate stability of equilibrium points and search for region(s) of stability in the parameter space of a general mathematical model with delay of two species. Model assumptions were developed based on the biological type of interaction called competition of two species.

Section snippets

The mathematical model

It is well known that a general mathematical model of two interacting species can be described by system of equations of the form, ${\begin{matrix} \frac{{\dot{N}}_{1}}{N_{1}} = F (N_{1}, N_{2}), \\ \frac{{\dot{N}}_{2}}{N_{2}} = G (N_{1}, N_{2}), \end{matrix}$ where N₁- population of species 1, N₂- population of species 2 and the functions F and G represent characteristics of the type of interactions between the species.

If mathematical model (1) is subjected to reproduction delays of the two species, then the model has a new form given by ${\begin{matrix} \frac{{\dot{N}}_{1} (t)}{N_{1} (t)} = F (N_{1} (t - τ_{1}), N_{2} (t - τ_{1})), \\ \frac{{\dot{N}}_{2} (t)}{N_{2} (t)} = G (N_{1} (t - τ_{2}) \end{matrix}$

Theory

In this section, discussion on d-decomposition method related to our problem of investigation is made based on the literature [7,12].

We investigated the influence of two parameters on the stability of the equilibrium points the dynamic system (2) that have a linear relationship with characteristic equation of the dynamic system given in the form $D (λ) = v N (λ) + μ M (λ) + L (λ) = 0 .$ Substituting λ = iω in Eq. (3) we obtain $D (i ω) = v N (i ω) + μ M (i ω) + L (i ω) = 0,$ and upon introducing, ${\begin{matrix} N (i ω) = : N_{1} (ω) + i N_{2} (ω), \\ M (i ω) = : M_{1} (ω) + i M_{2} (ω) \end{matrix}$

Result

In this section, local stability analysis of the marginal equilibrium points p₂ and p₄ and that of the interior equilibrium point p₃ is made.

Discussion

To verify the applicability of the result obtained for the general mathematical model, we considered a particular mathematical model for a mixed trophy and warfare competition type given below [14]. ${\begin{matrix} \frac{{\dot{N}}_{1}}{N_{1}} = F (N_{1}, N_{2}) = \frac{r_{1} (K_{1} - N_{1} - a N_{2})}{K_{1}} - c N_{2}, \\ \frac{{\dot{N}}_{2}}{N_{2}} = G (N_{1}, N_{2}) = \frac{r_{2} (K_{2} - N_{2} - b N_{1})}{K_{2}} - d N_{1}, \end{matrix}$ where, $\begin{matrix} N_{1}, N_{2} - actual population densities, \\ K_{1}, K_{2} - maximum population densities in the absence of competitors, \\ r_{1}, r_{2} - popaulation growth rates, \\ b, a - per cpita trophic pre - empting competion coeffcients, \\ d, c - per capita killing competition coefficient . \end{matrix}$

All the

Conclusion

The result of this research can be used to obtain parameter values that ensure the interior equilibrium point of any specific mathematical model representing competition of two species in a given ecosystem locally asymptotically stable. Once such parameter values are identified then it can be used to design a management or a control system of the ecosystem. Moreover, the method of the research can be adapted to other type of biological interactions, by adjusting the model assumptions.

Acknowledgement

This work has been supported by Jimma University, college of natural sciences staff research fund, Ethiopia .

References (14)

J. Wei et al.
Stability and bifurcation in a neural network model with two delays
Physica D
(1999)
K.M. Weiss
A Generalized Model of Computation between Hominids Population
J. Hum. Evol.
(1972)
K.L. Cooke et al.
Analysis of an SEIRS epidemic model with two delays
J. Math. Biol.
(1996)
H.I. Freedman et al.
Stability criteria for a system involving two time delays
SIAMJ. Appl. Math.
(1986)
S. Ruan et al.
On the zeros of transcendental functions with applications to stability of delay deferential equations with two delays
Dynam. Contain, Discrete Impulse Systems: Series A: Math. Anal.
(2003)
G. Stepan
Retarded Dynamical Systems: Stability and Characteristic Functions
(1989)
E.P. Popov
The Dynamics of Automatic Control Systems
(1962)

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Parameter based stability analysis of generalized mathematical model with delay of competition between two species

Highlights

Abstract

Introduction

Section snippets

The mathematical model

Theory

Result

Discussion

Conclusion

Acknowledgement

Physica D

J. Hum. Evol.

Analysis of an SEIRS epidemic model with two delays

J. Math. Biol.

Stability criteria for a system involving two time delays

SIAMJ. Appl. Math.

On the zeros of transcendental functions with applications to stability of delay deferential equations with two delays

Dynam. Contain, Discrete Impulse Systems: Series A: Math. Anal.

Retarded Dynamical Systems: Stability and Characteristic Functions

The Dynamics of Automatic Control Systems