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Title: | Dynamics of a Single Species under Periodic Habitat Fluctuations and Allee Effect |
Authors: | Rizaner, Fatma Bayramoğlu |
Keywords: | Mathematics Differential Equations, Nonlinear Nonlinear Differential Equation - Allee Effect - Periodic Solutions - Stability - Blow Up - Existence - Positive Solutions - Harvesting |
Issue Date: | 2012 |
Publisher: | Eastern Mediterranean University (EMU) |
Citation: | Rizaner, Fatma Bayramoglu. (2012). Dynamics of a Single Species under Periodic Habitat Fluctuations and Allee Effect. Thesis (Ph.D.), Eastern Mediterranean University, Institute of Graduate Studies and Research, Dept. of Mathematics, Famagusta: North Cyprus. |
Abstract: | The dynamics of a single species and harvested single species that goes extinct when rare, is described by nonlinear differential equations a)Ń = rN ﴾1 – N/K﴿ ﴾N/K – A/K﴿ b)Ń = rN ﴾1 – N/K﴿ ﴾N/K – A/K﴿-hN, (1) where a parameter A(0<A<K) is associated with the Allee effect, r is the
intrinsic growth rate, h is the harvesting and K is the carrying capacity of the
environment. The intention of this thesis is to study the existence of periodic
solutions and their stability properties assuming that r, A, h and K are continuous
T - periodic functions. Using rather elementary techniques, we completely describe
populations dynamics analyzing influences of both strong (A>0)and weak (A<0)Allee effects. We discuss the effect of harvesting on a single species population in a
fluctuating environment whose dynamics is described by a nonlinear differential
equation. We consider separately cases of harvesting (h>0)(stocking (h<0)), weak Allee effect (A≤0)and strong Allee effect (A>0). Thus, we answer
questions regarding the location of positive periodic solutions and their stability
complementing the research in a recent paper by Padhi [14]. Bounds for periodic
solutions and estimates for backward blow-up times are also established.
Furthermore, we demonstrate advantages of our approach on simple examples to
which the results in the cited paper fail to apply. Keywords: Nonlinear differential equation, Allee effect, periodic solutions, stability, blow up, existence, positive solutions, harvesting. …………………………………………………………………………………………………………………………………………………………………………………………………………………… ÖZ: Yetersiz nüfus yogunlugundan dolayı soyu tükenmekte olan tek bir türün ve hasat edilen tek bir türün dinamikleri dogrusal olmayan asagıdaki diferansiyel denklemlerle tanımlanabilir, a)Ń = rN ﴾1 – N/K﴿ ﴾N/K – A/K﴿ b)Ń = rN ﴾1 – N/K﴿ ﴾N/K – A/K﴿-hN, (1) Burada, A parametresi A(0<A<K) Allee etkisi ile iliskilidir, r içsel büyüme oranı, h hasat kaldırma ve K çevrenin tasıma kapasitesidir. Bu tezin amacı r, A, h ve K’nin sürekli T - periyodik fonksiyonlar oldukları kosullarda, periyodik çözümlerin varlıgını ve onların denge özelliklerini arastırmaktır. Temel teknikler kullanarak güçlü (A>0) ve zayıf (A<0) Allee etkileri incelenerek nüfus dinamikleri tamamıyla elde edilmislerdir. Dinamikleri dogrusal olmayan diferansiyel denklemlerle tanımlanan dalgalanma ortamındaki tek bir nüfusun hasatı incelenmemistir. Bu durumda ayrı ayrı hasat (h>0) (stok (h<0)), zayıf Allee etkisi (A≤0) ve güçlü Allee etkisi (A>0) dikkate alınmıştır. Bu arastırmayla Padhi’nin makalesinde [14] ortaya çıkan pozitif periyodik çözümlerin konumları ve bunların istikrarlarıyla ilgili soruları aydınlattık. Periyodik çözümlerin sınırları ve geri darbe süreleri de tanımlanmıstır. Ayrıca, bu çalısmada önerilen yaklasımın avantajı Padhi’nin makalesinde [14] önerdigi sonuçların uygulanmayacagı basit örnekler yardımıyla gösterilmistir. |
Description: | Doctor of Philosophy in Mathematics. Thesis (Ph.D.)--Eastern Mediterranean University, Faculty of Arts and Sciences, Dept. of Mathematics, 2012. Supervisor: Assoc. Prof. Dr. Svitlana P. Rogovchenko. |
URI: | http://hdl.handle.net/11129/350 |
Appears in Collections: | Theses (Master's and Ph.D) – Mathematics
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