• Kyungpook Mathematical Journal
• /
• 제57권2호
• /
• pp.265-285
• /
• 2017

In this article, a mathematical model is proposed and analyzed to study the delayed dynamics of a system having a predator and two preys with distinct growth rates and functional responses. The equilibrium points of proposed system are determined and the local stability at each of the possible equilibrium points is investigated by its corresponding characteristic equation. The boundedness of the system is established in the absence of delay and the condition for existence of persistence in the system is determined. The discrete type gestational delay of predator is also incorporated on the system. Further it is proved that the system undergoes Hopf bifurcation using delay as bifurcation parameter. This study refers that time delay may have an impact on the stability of the system. Finally Computer simulations illustrate the dynamics of the system.

• 대한수학회지
• /
• 제55권1호
• /
• pp.43-58
• /
• 2018

In this paper, we study a delayed predator-prey system with Holling type IV functional response incorporating the effect of habitat complexity. The results show that there exist stability switches and Hopf bifurcation occurs while the delay crosses a set of critical values. The explicit formulas which determine the direction and stability of Hopf bifurcation are obtained by the normal form theory and the center manifold theorem.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제13권1호
• /
• pp.21-29
• /
• 2009

This paper treats the conditions for the existence and stability properties of stationary solutions of a predator-prey interaction with self and cross-diffusion. We show that at a certain critical value a diffusion driven instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing instability takes place. We note that the cross-diffusion increase or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges.

• Kyungpook Mathematical Journal
• /
• 제49권4호
• /
• pp.605-618
• /
• 2009

A stage-structured predator-prey system with time delay and Beddington-DeAngelis functional response is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated. The existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

• 대한수학회논문집
• /
• 제21권2호
• /
• pp.293-320
• /
• 2006

Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.

• Journal of applied mathematics & informatics
• /
• 제28권5_6호
• /
• pp.1379-1393
• /
• 2010

In this paper, a stage-structured predator prey model (stage structure on prey) with two discrete time delays has been discussed. The two discrete time delays occur due to maturation delay and gestation delay. Linear stability analysis for both non-delay as well as with delays reveals that certain thresholds have to be maintained for coexistence. Numerical simulation shows that the system exhibits Hopf bifurcation, resulting in a stable limit cycle.

• Kyungpook Mathematical Journal
• /
• 제50권2호
• /
• pp.255-266
• /
• 2010

We consider a delayed predator-prey system with Holling II functional response. Firstly, the paper considers the stability and local Hopf bifurcation for a delayed prey-predator model using the basic theorem on zeros of general transcendental function, which was established by Cook etc.. Secondly, special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are given.

• Journal of applied mathematics & informatics
• /
• 제10권1_2호
• /
• pp.17-26
• /
• 2002

A stability analysis of a non-linear prey-predator system under the influence of one dimensional diffusion has been investigated to determine the nature of the bifurcation point of the system. The non-linear bifurcation analysis determining the steady state solution beyond the critical point enables us to determine characteristic features of the spatial inhomogeneous pattern arising out of the bifurcation of the state of the system.

• 한국컴퓨터정보학회논문지
• /
• 제26권9호
• /
• pp.27-35
• /
• 2021

인공생명체 연구는 자연 생명과 관련된 시스템이나 그 과정들, 진화 등을 평가해 다양한 응용과학 분야에 활용된다. 이러한 인공생명체의 원활한 활동을 위해 물리적 신체 설계와 행동 제어전략을 진화시키는 연구가 활발히 진행되었다. 그러나 형태와 신경망을 공진화시키는 것은 어렵기에 최적화된 움직임을 가진 인공생명체는 한 가지 형태에 한 가지 움직임만을 가지며 주변 환경 상황은 고려하지 않는 것이 대부분이다. 본 논문에서는 포식자-피식자 모델을 이용하여 형태와 신경망을 공진화하는 인공생명체가 환경적응형 움직임을 갖게 한다. 그런 다음 포식자-피식자 계층 구조를 최상위 포식자-중간 포식자-최하위 피식자 3단계로 확장하여 초기 개체군 밀도에 따라 시뮬레이션의 안정성을 판별하며 형태 진화와 개체군 역학 간의 상관관계를 분석한다.

• 한국시뮬레이션학회논문지
• /
• 제22권3호
• /
• pp.1-6
• /
• 2013

생태계는 다양한 환경 내에 다양한 생물종이 서로 상호작용하고 있는 복잡계이다. 이들 상호작용은 계층적 먹이그물 구조를 이루고 있는데, 많은 경우, 포식자-피식자-식물의 관계를 보여준다. 포식자-피식자 경쟁관계는 시공간적으로 일어나는 현상이기 때문에, 초기시점에서의 개체들 분포와 밀도가 어떠한가는 매우 중요한 정보를 담고 있다. 본 연구에서는, 이들 세 단계 계층구조의 생태계를 간단한 격자 모델로 구성하고 이 모델을 사용하여 각 종의 초기 개체군 밀도가 변함에 따라 생태계 안정성이 어떻게 변하는지를 연구하였다. 격자공간은 $L{\times}L$ 크기의 L(=100) 사각격자로 구성되었다. 식물의 초기 밀도는 0.2로 고정하였다. 시뮬레이션 결과는, 포식자의 밀도가 0.4이하, 피식자의 밀도가 0.5이하일 때 두 종이 공존하는 것을 보여 주었으며, 포식자 밀도가 0.5이상, 피식자 밀도가 0.6 이상의 조건에서는 두 종이 멸종하는 것을 보여 주었다. 공존과 멸종의 두 상태가 접하는 영역의 조건에서는 확률적으로 공존하기도하고 멸종하기도 하는 비선형성이 강한 행동을 보여 주었다. 본 연구를 통해 초기종의 밀도가 생태계 안정성에 매우 중요한 역할을 한다는 것을 알 수 있었다.