蒲武軍，杜爭光

一類分?jǐn)?shù)階比率依賴型捕食系統(tǒng)的動力學(xué)分析

*蒲武軍，杜爭光

（隴南師范高等?？茖W(xué)校數(shù)學(xué)系，甘肅，隴南 742500）

討論了一類食餌帶有疾病的分?jǐn)?shù)階比率依賴型捕食系統(tǒng)的動力學(xué)行為。利用線性化方法定性分析了各類平衡點(diǎn)的穩(wěn)定性，并給出了其正平衡點(diǎn)局部漸近穩(wěn)定的充分條件。數(shù)值模擬顯示，參數(shù)和階數(shù)對平衡點(diǎn)的收斂速度及其穩(wěn)定性產(chǎn)生很重要的影響。

分?jǐn)?shù)階；比率依賴型捕食系統(tǒng)；局部漸近穩(wěn)定性；Hopf分支

近年來，隨著計(jì)算機(jī)技術(shù)的不斷發(fā)展，分?jǐn)?shù)階計(jì)算引起了許多學(xué)者的關(guān)注，并成功應(yīng)用到工程技術(shù)領(lǐng)域[1-2]，特別是許多應(yīng)用數(shù)學(xué)工作者[3-7]用分?jǐn)?shù)階計(jì)算模擬現(xiàn)實(shí)過程。靳楨[8]研究了一類食餌帶有疾病的比率依賴型捕食系統(tǒng)，討論了其平衡點(diǎn)的穩(wěn)定性。受此啟發(fā)，本文考慮如下分?jǐn)?shù)階模型：

1 預(yù)備知識

引理1[10]考慮如下的自治系統(tǒng)：

引理2[11]考慮如下的分?jǐn)?shù)階系統(tǒng)：

2 主要結(jié)果及其證明

2.1 解的存在唯一性

2.2 平衡點(diǎn)的局部穩(wěn)定性

情形1：

情形2：

方程(8)三個特征根可表示為

其中

證明：結(jié)論(i)和(iii)顯然成立，故只需證明結(jié)論(ii)成立即可，注意到

3 數(shù)值模擬

圖1 易感者S，感染者I，捕食者R的收斂性及其相圖

圖2 不同參數(shù)下易感者S，感染者I，捕食者R的收斂性及其相圖

圖3 α=0.988時，易感者S，感染者I，捕食者R的振蕩及正平衡點(diǎn)附近的周期性閉軌

圖4 α=0.988時，易感者S，感染者I，捕食者R發(fā)散及正平衡點(diǎn)不穩(wěn)定

4 討論

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[5] 劉娜,方潔,鄧瑋,等. 一類分?jǐn)?shù)階復(fù)值SIR傳染病模型的穩(wěn)定性分析[J].數(shù)學(xué)的實(shí)踐與認(rèn)識,2019,23:256-261.

[6] 趙新俊,鄭義. 分?jǐn)?shù)階SIR傳染病模型的穩(wěn)定性分析[J]. 生物數(shù)學(xué)學(xué)報,2018,1:11-16.

[7] 杜爭光. 具有Holling Ⅳ型功能反應(yīng)的分?jǐn)?shù)階捕食者-食餌模型的動力學(xué)分析[J].井岡山大學(xué)學(xué)報:自然科學(xué)版,2019,40(3):9-13.

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DYNAMICS OF A RATIO-DEPENDENT PREDATOR-PREY OF FRACTIONAL ORDER SYSTEM

*PU Wu-jun, DU Zheng-guang

(Department of Mathematics, Longnan Teachers College, Longnan, Gansu 742500, China)

In this paper, we studied the dynamic behavior of a fractional-order ratio-dependent predator–prey system with disease on the prey. The stability of all kinds of equilibrium points of the system was qualitatively analyzed with the linearization method, the sufficient conditions for the local asymptotic stability of the positive equilibrium point of the system were given, and the numerical simulation showed that the parameters and order of the system had an important influence on the convergence rate and stability of the equilibrium point.

fractional-order; ratio-dependent predator-prey model; local asymptotic stability; Hopf bifurcation

O157.13

A

10.3969/j.issn.1674-8085.2021.02.003

1674-8085(2021)02-0014-05

2020-10-14；

2020-12-04

隴南市2019年科技指導(dǎo)性計(jì)劃項(xiàng)目（2019-ZD-14）

*蒲武軍(1979-)，男，甘肅莊浪人，講師，碩士，主要從事生物數(shù)學(xué)方面的研究(E-mail:puwj2005@163.com);

杜爭光(1973-)，男，甘肅禮縣人，副教授，主要從事應(yīng)用微分方程方面的研究(E-mail:lnsz_dzg@163.com).