Abstract
Incorporating two delays (tau_{1} represents the maturity of predator, tau_{2} represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov–LaSalle’s principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if tau_{1}in (m_{1},m_{2}) and tau_{2}in(m_{4}, +infty), then the boundary equilibrium E_{2}(x^{0}, y_{1}^{0}, y_{2}^{0}, 0, 0) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if tau_{1}in(m_{2}, +infty), then the axial equilibrium E_{1}(k, 0, 0, 0, 0) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results.
Highlights
Predator–prey type interaction is one of basic interspecies relations in the biology and ecology and it is the basic block of the complicated food chain, food web and biochemical network structure [1,2,3,4]
4 Local stability analysis of the equilibria we study the local stability of system (2) at equilibria
As for this problem, we investigate the stability of the interior equilibrium E∗ by constructing a suitable Lyapunov functional and applying Lyapunov–LaSalle’s principle
Summary
Predator–prey type interaction is one of basic interspecies relations in the biology and ecology and it is the basic block of the complicated food chain, food web and biochemical network structure [1,2,3,4]. Remark 2 Since we consider a three-species-food-chain model, the dynamical behaviors are more complicated and the system has more equilibria than those in [4, 10, 12] These conditions of (C2), (C3) and (C4) seem to be intricate, take Case I When deal with the distribution of characteristic roots for the transcendental equation like λ3 + cλ2 + a1λ + a2 + (b1λ + b2)e–λτ1 = 0, the local stability of the interior equilibrium E∗ cannot be derived by Lemma 3.1 [8]. That is novel and different from [8, 10, 14, 24]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
