422 research outputs found
Stability Analysis of a Ratio-Dependent Predator-Prey Model Incorporating a Prey Refuge
A ratio-dependent predator-prey model incorporating a prey refuge with disease in the prey population is formulated and analyzed. The effects of time delay due to the gestation of the predator and stage structure for the predator on the dynamics of the system are concerned. By analyzing the corresponding characteristic equations, the local stability of a predator-extinction equilibrium and a coexistence equilibrium of the system is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the coexistence equilibrium, when Ď„=Ď„0. By comparison arguments, sufficient conditions are obtained for the global stability of the predator-extinction equilibrium. By using an iteration technique, sufficient conditions are derived for the global attractivity of the coexistence equilibrium of the proposed system
Stability and Bifurcation Analysis on an Ecoepidemiological Model with Stage Structure and Time Delay
An ecoepidemiological predator-prey model with stage structure for the predator and time delay due to the gestation of the predator is investigated. The effects of a prey refuge with disease in the prey population are concerned. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the model is discussed. Further, it is proved that the model undergoes a Hopf bifurcation at the positive equilibrium. By means of appropriate Lyapunov functions and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the semitrivial boundary equilibria. By using an iteration technique, sufficient conditions are derived for the global attractiveness of the positive equilibrium
Theoretical Study of Pest Control Using Stage Structured Natural Enemies with Maturation Delay: A Crop-Pest-Natural Enemy Model
In the natural world, there are many insect species whose individual members
have a life history that takes them through two stages, immature and mature.
Moreover, the rates of survival, development, and reproduction almost always
depend on age, size, or development stage. Keeping this in mind, in this paper,
a three species crop-pest-natural enemy food chain model with two stages for
natural enemies is investigated. Using characteristic equations, a set of
sufficient conditions for local asymptotic stability of all the feasible
equilibria is obtained. Moreover, using approach as in (Beretta and Kuang,
2002), the possibility of the existence of a Hopf bifurcation for the interior
equilibrium with respect to maturation delay is explored, which shows that the
maturation delay plays an important role in the dynamical behavior of three
species system. Also obtain some threshold values of maturation delay for the
stability-switching of the particular system. In succession, using the normal
form theory and center manifold argument, we derive the explicit formulas which
determine the stability and direction of bifurcating periodic solutions.
Finally, a numerical simulation for supporting the theoretical analysis is
given.Comment: 28 pages, 9 figure
Predator-prey dynamics pertaining to structuralizing predator species into three stages coupled with maturation delay owing to juvenile hunting
The predator-prey dynamic appertaining to two species is explored, wherein
the predator species is structured into different stages. As evidenced from
natural documentation, the immature predators possess the potential to predate
albeit not as competently as the adults. Nevertheless, this potentiality is not
acquired immediately after their incipience of life, hence, the immature stage
is branched off into the infant stage, the stage with extensive reliance on the
adults, and the juvenile stage, the stage with the potential to predate but not
to procreate. In this paper, this inaugural concept is coupled with injuries in
the juvenile stage as the repercussion of their incompetency in predating,
thereby ensuing a delay in their maturation. With the incentive to investigate
the ascendancy of these refinements over the whole system, stability analyses
along with various bifurcation analyses around the equilibrium points of the
system are corroborated. In addition to Hopf, transcritical, and saddle node
bifurcations, the existence of Bogdanov-Takens point, cusp point, Bautin
bifurcation point, bloom phenomenon, twice occurring Hopf bifurcation, and
bi-stability phenomenon make the paper appreciably more rich and efficacious
Bifurcation analysis for a delayed food chain system with two functional responses
A delayed three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is investigated. By analyzing the distribution of the roots of the associated characteristic equation, we get the sufficient conditions for the stability of the positive equilibrium and the existence of Hopf bifurcation. In particular, using the normal form theory and center manifold theorem, the properties of Hopf bifurcation such as direction and stability are determined. Finally, numerical simulations are given to substantiate the theoretical results
Controllability of an eco-epidemiological system with disease transmission delay: A theoretical study
This paper deals with the qualitative analysis of a disease transmission delay induced prey preda-tor system in which disease spreads among the predator species only. The growth of the preda-tors’ susceptible and infected subpopulations is assumed as modified Leslie–Gower type. Suffi-cient conditions for the persistence, permanence, existence and stability of equilibrium points are obtained. Global asymptotic stability of the system is investigated around the coexisting equilib-rium using a geometric approach. The existence of Hopf bifurcation phenomenon is also exam-ined with respect to some important parameters of the system. The criterion for disease a trans-mission delay the induced Hopf bifurcation phenomenon is obtained and subsequently, we use a normal form method and the center manifold theorem to examine the nature of the Hopf bifurca-tion. It is clearly observed that competition among predators can drive the system to a stable from an unstable state. Also the infection and competition among predator population enhance the availability of prey for harvesting when their values are high. Finally, some numerical simu-lations are carried out to illustrate the analytical results
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