Effects of Diffusion and Delays on the Dynamic Behavior of a Competition and Cooperation Model

Hassan Y. Alfifi ()
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Hassan Y. Alfifi: Department of General Courses, College of Applied Studies and Community Service, Imam Abdulrahman Bin Faisal University, Dammam 34211, Saudi Arabia

Mathematics, 2025, vol. 13, issue 7, 1-21

Abstract: This study investigates a model of competition and cooperation between two enterprises with reaction, diffusion, and delays. The stability and Hopf bifurcation for variants with two, one, and no delays are considered by examining a system of delay ODE equations analytically and numerically, applying the Galerkin method. A condition is obtained that helps characterize the existence of Hopf bifurcation points. Full maps of stability analysis are discussed in detail. With bifurcation diagrams, three different cases of delay are shown to determine the stable and unstable regions. It is found that when τ i > 0 , there are two different stability regions, and that without a delay ( τ i = 0 ), there is only one stable region. Furthermore, the effects of delays and diffusion parameters on all other free rates in the system are considered; these can significantly affect the stability areas, with important economic consequences for the development of enterprises. Moreover, the relationship between the diffusion and delay parameters is discussed in more detail: it is found that the value of the time delay at the Hopf point increases exponentially with the diffusion coefficient. An increase in the diffusion coefficient can also lead to an increase in the Hopf-point values of the intrinsic growth rates. Finally, bifurcation diagrams are used to identify specific instances of limit cycles, and 2-D phase portraits for both systems are presented to validate all theoretical results discussed in this work.

Keywords: delay; stability; bifurcation; reaction–diffusion; periodic solutions; enterprise; competition; cooperation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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