PROC
On certain elliptic systems with nonlinear self-cross diffusions
Kimun Ryu Inkyung Ahn
We investigate the positive coexistence to certain strongly-coupled nonlinear elliptic systems with self-cross diffusions under homogeneous Robin boundary conditions. Competing interactions between two species are considered. Conditions of the positive coexistence to self-cross diffusive systems can be expressed in terms of the spectral property of differential operators of nonlinear Schrödinger type which reflect the influence of the domain and nonlinearity in the system. Decoupling method and nonlinear fixed point theorem are employed.
keywords: decomposing operator. positive coexistence nonlinear elliptic system fixed point index nonlinear self-cross diffusions
DCDS
Positive steady--states for two interacting species models with linear self-cross diffusions
Kimun Ryu Inkyung Ahn
In this paper, we discuss the positive steady-state existence for predator-prey and competing interaction systems between two species with linear self-cross diffusions. The methods employed are the decomposing operators and the theory of fixed point index on cones in a Banach space. We give sufficient conditions for the existence of positive solutions. The conditions are given in terms of the signs of the principal eigenvalues of certain differential operators.
keywords: fixed point index. linear self-cross diffusion Elliptic system positive coexistence decomposing operator
CPAA
Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction
Wonlyul Ko Inkyung Ahn

We consider a predator-prey system with a ratio-dependent functional response when a prey population is infected. First, we examine the global attractor and persistence properties of the time-dependent system. The existence of nonconstant positive steady-states are studied under Neumann boundary conditions in terms of the diffusion effect; namely, pattern formations, arising from diffusion-driven instability, are investigated. A comparison principle for the parabolic problem and the Leray-Schauder index theory are employed for analysis.

keywords: Non-constant positive steady states eco-epidemiological model predator-prey interaction global attractor pattern formation
CPAA
Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses
Seong Lee Inkyung Ahn

In this paper, we examine a diffusive predator-prey model with Beddington-DeAngelis functional response and stage structure on prey under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature prey to their maturity. We investigate the dynamics of their permanence and the extinction of the predator, and provide sufficient conditions for the global attractiveness and the locally asymptotical stability of the semi-trivial and coexistence equilibria.

keywords: Diffusive predator-prey model Beddington-DeAngelis functional response time delay stage structure on prey locally/globally asymptotically stable
PROC
Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey
Inkyung Ahn Wonlyul Ko Kimun Ryu
In this paper, we consider a ratio-dependent predator-prey model with disease in the prey under Neumann boundary condition. we construct a global attractor region for all time-dependent non-negative solutions of the system and investigate the asymptotic behavior of positive constant solution. Furthermore, we also study the asymptotic behavior of the non-negative equilibria.
keywords: predator-prey disease-free asymptotic stability. Ratio-dependent
DCDS-B
Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay
Wonlyul Ko Inkyung Ahn Shengqiang Liu
In this paper, we examine the asymptotic behaviors of a diffusive delayed consumer-resource model subject to homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of juvenile consumers to their maturity, and the predation is of a general type of functional response. We construct the threshold dynamics of the persistence and extinction of the consumer. Moreover, we establish the sufficient conditions for the global attractivity of the semitrivial and coexistence equilibria. Finally, we apply our results to the specific consumer-resource models with Beddington-DeAngelis, Crowley-Martin, and ratio-dependent type of functional responses.
keywords: permanence global stability Beddington-DeAngelis/Crowley-Martin/ratio-dependent models. Lyapunov function general functional response maturation delay Consumer-resource model

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