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Example 1: Consider the following predator-prey system
饮水机消毒The solution curves in the pha plane is as follows:
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1
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Sketch the R(t)- and F(t)-graphs for the solutions with initial points A, B, and C.
2
Find the longterm behavior of the populations over time.
灌肠步骤Answer: First, notice that the point A is an equilibrium point with non-zero coordinates. If we t
we get the following three points: (0,0), (2.5,0), and (1/0.8,1/1.5). Clearly, we have
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1
The sketches of the R(t)- and F(t)- graphs for the solutions with initial points A, B, and C respectively, are given below
2
It is clear that all the solutions on the pha-plane tend to the equilibrium point A when t goes to infinity.
Therefore, in all the cas, we have
,
Example 2: The following depicts the pha portrait of the solution to an autonomous system of differential equations with initial conditions x(0)=2 and y(0)=1. Sketch the time ries for x(t) and y(t).
Answer:
Example 3: Consider the first order system
1.
Graph the nullclines for this system.
2.
Inrt direction field "arrows" at the nullclines
3.
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Determine the general direction of the direction field in each of the regions cut out by the nullclines. 4.
Sketch possible pha portraits for the two initial conditions indicated by "dots" for as long as the pha portraits stay within the graph's frame.
5.
Find all the equilibrium points of the system.
Answer:
1.
The x-nullclines can be obtained by tting x'=0, i.e.
The y-nullclines can be obtained by tting y'=0, i.e.
