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这是一篇英国作业为数学生物的限时测试
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），
其中a，b，c，d，e，J> 0。
（a）简要讨论该模型，确定物种-物种相互作用的类型
涉及。
（b）找到系统的所有稳态（1）并确定它们是否在本地
稳定或不稳定。
（c）当ae 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
不稳定。
（d）表明当Ka> 2fi时，内部稳定周围可能存在极限循环
随A的状态变化而求出A的临界值Ac。
（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.