Gecikme ve yaşam alanı karmaşıklığı eklenmiş av-avcı tipindeki etkileşimler için nümerik çatallanma analizi

Bu makalede avcı türü reaksiyonuna yaşam alanı karmaşıklığı ve olgunlaşma evresine sabit bir gecikme eklenerek elde edilen genel bir iki bileşenli av-avcı sistemi dikkate alınmıştır. Zaman gecikmesinin sistem dinamiklerine etkisi geniş bir şekilde literatürde çalışılmış olmasına ragmen, sadece birkaç araştırmacı av-avcı tipinde etkileşimlerde yaşam alanı karmaşıklığının etkilerini ele almıştır. Makalenin ilk bölümünde denge noktaları ve matematiksel modelin kararlılık analizinden bahsedilmiştir. İkinci kısımda av ve avcı türlerinin yoğunluklarının gecikmeli ve gecikmesiz durumlarda yapılan nümerik çatallanma analizine iki parametreye bağlı olarak dikkat çekilmiştir:(i) homojen yaşam alanı karmaşıklığının etkinlik parametresi ve (ii) avcı saldırı oranı parametresi. Gecikmesiz sistemde dinamiklerin kararlılığının Hopf çatallanması ile kararlı durumdan kararsız duruma değiştiği ve bu Hopf noktalarından yayılan dalların genelde kararlı ve süper kritik olduğu gözlemlenmiştir. Buna rağmen, gecikme kaynaklı sistem ise Hopf çatallanmadan çıkan kararsız yörüngelere sebep olabilir. Aynı zamanda yaşam alanı karmaşıklığının etkinliğini artırmanın sistem dinamiklerinin kararsız durumdan kararlı duruma geçmesine sebep olduğu bulunmuştur.

Numerical bifurcation analysis for a prey-predator type interactions with a time lag and habitat complexity

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predatorfunctional response, and with constant time delay in predator gestation is considered. Although the role of timedelay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effectof habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stabilityanalysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numericalbifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength ofhomogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamicswith time delay in predator gestation are found to be much richer compared to that without time delay. The systemstability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating fromthese Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbitsarising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead thestability change from unstable to stable.

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