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1.系统模拟了两个相互作用的物种，其密度为x和y
— = x {a-bx-cy），
f（1）ay，„，
— = y（-d + ex-fy），

（a）简要讨论该模型，确定物种-物种相互作用的类型

（b）找到系统的所有稳态（1）并确定它们是否在本地

（c）当ae <bd时，绘制系统（1）的相平面。
（d）描述当6 = 0 = /时捕食者密度可能的时间演变。
2.捕食者－被捕食者模型具有以下形式
dt \ i \ / ^ \

7 Wi / Vo其中0（iVj，N2）= -r1-rr和p，K，ry，A，o-均为正常数。 A-\-yVj
（a）N 1和N 2中的哪个代表掠食者，哪个猎物？
（b）对于固定的N2> 0，草图4>（Ni> ^ 2）。参数7代表什么？
（c）找到（2）的所有稳态，并确定它们是否是局部稳定的或

（d）表明当Ka> 2fi时，内部稳定周围可能存在极限循环

（e）当A小于Ac时，画出（2）的相平面

1. Two interacting species with densities x and y are modelled by the system
— = x{a-bx-cy),
f (1) ay , , „ ,
— = y(-d + ex-fy),
where a, b, c, d,e,J > 0.
(a) Briefly discuss the model, identifying the type of species-species interactions
involved.
(b) Find all steady states of the system (1) and determine whether they are locally
stable or unstable.
(c) Sketch the phase planes for the system (1) when ae < bd.
(d) Describe the possible time evolutions of the predator density when 6 = 0 = /.
2. A predator-prey model has the form
dt \ i\ / ^\
i
7 Wi /Vo where 0(iVj,N2) = -r1—rr and p,K,ry,A,o- are all positive constants. A -\- yVj
(a) Which of N\ and N2 represents the predator, and which the prey?
(b) Sketch 4>(Ni> ^2) for a fixed N2 > 0. What does the parameter 7 represent?
(c) Find all steady states of (2) and determine whether they are locally stable or
unstable.
(d) Show that when Ka > 2fi, a limit cycle is possible around the interior steady
state as A varies and find the critical value Ac of A at which it occurs.
(e) Sketch the phase plane for (2) when A is just less than Ac.

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