Phytoplanktons are drifting plants in an aquatic system.They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis.Zooplanktons are drifting ...Phytoplanktons are drifting plants in an aquatic system.They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis.Zooplanktons are drifting animals found inside the aquatic bodies.For stable aquatic ecosystem,the growth of both Zooplankton and Phytoplankton should be in steady state but in previous eras,there has been a universal explosion in destructive Plankton or algal blooms.Many investigators used various mathematical methodologies to try to explain the bloom phenomenon.So,in this paper,a discretized two-dimensional Phytoplankton-Zooplankton model is investigated.The results for the existence and uniqueness,and conditions for local stability with topological classifications of the equilibrium solutions are determined.It is also exhibited that at trivial and semitrivial equilibrium solutions,discrete model does not undergo fip bifurcation,but it undergoes Neimark-Sacker bifurcation at interior equilibrium solution under certain conditions.Further,state feedback method is deployed to control the chaos in the under consideration system.The extensive numerical simulations are provided to demonstrate theoretical results.展开更多
In this papcr,bifurcations and chaos control in a discrete-time Lotka-Volterra predator-prey model have been studied in quadrant-I.It is shown that for all parametric values,model hus boundary equilibria:P00(0,0),Px0(...In this papcr,bifurcations and chaos control in a discrete-time Lotka-Volterra predator-prey model have been studied in quadrant-I.It is shown that for all parametric values,model hus boundary equilibria:P00(0,0),Px0(1,0),and the unique positive equilibrium point:P^+xy(d/c,r(c-d)/bc) if c>d.By Linearization method,we explored the local dynamics along with different topological classifications about equilibria.We also explored the boundedness of positive solution,global dynamics,and existence of prime-period and periodic points of the model.It is explored that flip bifurcation occurs about boundary equilibria:Poo(0,0),P.o(1,0),and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of P^+xy(d/c,r(c-d)/bc).Further,it is also explored that about P^+xy(d/c,r(c-d)/bc) the model undergoes a N-S bifurcation,and meanwhile a stable close invariant curves appears.From the perspective of biology,these curves imply that betwecn predator and prey populations,there exist periodic or quasi-periodic oscillations.Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23.The Maximum Lyapunov exponents as well as fractal dimension are computed numeri-cally to justify the chaotic behaviors in the model.Finally,feedback control method is applied to stabilize chaos existing in the model.展开更多
基金The research of A.Q.Khan and F.Nazir is partially supported by the Higher Education Commission of Pakistan.
文摘Phytoplanktons are drifting plants in an aquatic system.They provide food for marine animals and are compared to terrestrial plants in that having chlorophyll and carrying out photosynthesis.Zooplanktons are drifting animals found inside the aquatic bodies.For stable aquatic ecosystem,the growth of both Zooplankton and Phytoplankton should be in steady state but in previous eras,there has been a universal explosion in destructive Plankton or algal blooms.Many investigators used various mathematical methodologies to try to explain the bloom phenomenon.So,in this paper,a discretized two-dimensional Phytoplankton-Zooplankton model is investigated.The results for the existence and uniqueness,and conditions for local stability with topological classifications of the equilibrium solutions are determined.It is also exhibited that at trivial and semitrivial equilibrium solutions,discrete model does not undergo fip bifurcation,but it undergoes Neimark-Sacker bifurcation at interior equilibrium solution under certain conditions.Further,state feedback method is deployed to control the chaos in the under consideration system.The extensive numerical simulations are provided to demonstrate theoretical results.
基金This work was supported by the Higher Education Cominission of Pakistan.
文摘In this papcr,bifurcations and chaos control in a discrete-time Lotka-Volterra predator-prey model have been studied in quadrant-I.It is shown that for all parametric values,model hus boundary equilibria:P00(0,0),Px0(1,0),and the unique positive equilibrium point:P^+xy(d/c,r(c-d)/bc) if c>d.By Linearization method,we explored the local dynamics along with different topological classifications about equilibria.We also explored the boundedness of positive solution,global dynamics,and existence of prime-period and periodic points of the model.It is explored that flip bifurcation occurs about boundary equilibria:Poo(0,0),P.o(1,0),and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of P^+xy(d/c,r(c-d)/bc).Further,it is also explored that about P^+xy(d/c,r(c-d)/bc) the model undergoes a N-S bifurcation,and meanwhile a stable close invariant curves appears.From the perspective of biology,these curves imply that betwecn predator and prey populations,there exist periodic or quasi-periodic oscillations.Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23.The Maximum Lyapunov exponents as well as fractal dimension are computed numeri-cally to justify the chaotic behaviors in the model.Finally,feedback control method is applied to stabilize chaos existing in the model.
