Using a procedure similar to that in the text for the...

Using a procedure similar to that in the text for the competitive hunter model, show that each trajectory is traversed in a counterclockwise direction as time $t$ increases. Along each trajectory, both the rabbit and fox populations fluctuate between their maximum and minimum levels. The maximum and minimum levels for the rabbit population occur where the trajectory intersects the horizontal line $y=a / b .$ For the fox population, they occur where the trajectory intersects the vertical line $x=c / d .$ When the rabbit population is at its maximum, the fox population is below its maximum value. As the rabbit population declines from this point in time, we move counterclockwise around the trajectory, and the fox population grows until it reaches its maximum value. At this point the rabbit population has declined to $x=c / d$ and is no longer at its peak value. We see that the fox population reaches its maximum value at a later time than the rabbits. The predator population lags behind that of the prey in achieving its maximum values. This lag effect is shown in Figure $9.38,$ which graphs both $x(t)$ and $y(t)$ .

Using a procedure similar to that in the text for the competitive hunter model, show that each trajectory is traversed in a counterclockwise direction as time $t$ increases.
Along each trajectory, both the rabbit and fox populations fluctuate
between their maximum and minimum levels. The maximum and
minimum levels for the rabbit population occur where the trajectory
intersects the horizontal line $y=a / b .$ For the fox population, they
occur where the trajectory intersects the vertical line $x=c / d .$ When
the rabbit population is at its maximum, the fox population is below
its maximum value. As the rabbit population declines from this point
in time, we move counterclockwise around the trajectory, and the fox
population grows until it reaches its maximum value. At this point the
rabbit population has declined to $x=c / d$ and is no longer at its peak
value. We see that the fox population reaches its maximum value at a
later time than the rabbits. The predator population lags behind that of
the prey in achieving its maximum values. This lag effect is shown in
Figure $9.38,$ which graphs both $x(t)$ and $y(t)$ .

Key Concepts

Phase Plane Analysis

Phase plane analysis involves studying the trajectories of a dynamical system in a space defined by its state variables. By plotting the populations of two interacting species against each other, one can visualize how the system evolves over time, including identifying fixed points, closed orbits, and the direction in which trajectories are traversed.

Predator-Prey Model

The predator-prey model is a mathematical framework used to describe the ecological interactions between predators and their prey, typically through a system of differential equations. It characterizes the growth and decline of populations, the effects of predation, and the resulting dynamic behavior in the system.

Trajectory Direction

Trajectory direction refers to the way in which the system's state moves around closed orbits or along paths in the phase plane. In the context of predator-prey models, a counterclockwise direction indicates a particular sequencing of events between the prey and predator populations as time progresses.

Population Oscillations

Population oscillations are the periodic fluctuations in the numbers of predators and prey around an equilibrium point. These oscillations typically arise from the intrinsic feedback interactions between the species, resulting in cycles of growth and decline that can be visualized as closed orbits in the phase plane.

Lag Effect

The lag effect is the phenomenon where the peak of one population, typically the predators, occurs after the peak of the other population, the prey. This delay in reaching a population maximum is a critical aspect of the dynamics in predator-prey models, highlighting the time-dependent response of one species to changes in the other.

Transcript

00:01 In this problem, we're given a very long prompt.

00:03 Using a procedure similar to competitive hunter model show that each trajectory is traversed in a counterclockwise direction as time the key is increased.

00:14 Along each trajectory, both the rabbit and fox populations fluctuate between their maximum and minimum values.

00:22 The maximum and minimum values for the rabbit population occur where the trajectory intersects the horizontal line, y equals a over b.

00:30 For the fox population, they occur where the trajectory intersects the vertical line, x equals c over d.

00:37 When the rabbit population is at its maximum, the fox population is below its maximum value.

00:43 As the rabbit population declines from this point in time, we move counterclockwise around the trajectory, and the fox population grows until it reaches its maximum value.

00:56 At this point, the rabbit population is declined to x equal c over d.

01:00 And is no longer at its peak value.

01:02 We see that the fox population reaches its maximum value at a later time, than the rabbits.

01:09 The predator population legs behind the prey and achieving maximum values.

01:13 This effect is shown right here, where red is the rabbits, and you can see that the fox population legs behind.

01:26 Okay, they go in the positive direction, and we've got in the figure lines a over b.

01:40 I've got to get to my problem again here.

01:49 Okay, x and y.

01:58 Okay.

02:08 So y equals a over b.

02:15 X equals c over d.

02:19 And we've got the minimum values.

02:21 Okay.

02:25 Both populations from the graph, periodic functions of time so that population of foxes at time equals zero is minimum.

03:08 So y -o equals, or is less than a over b, and x -o is equal to c over d.

03:28 And i'm going to put a couple lines in here.

03:40 Okay, it's possible to notice that moments after t is equal to t -0, both populations grow.

04:22 That's very apparent in the graph.

04:29 So d x over d t is greater than zero and d y over d t is greater than zero.

04:40 So the trajectory passes to region two.

05:00 And then after t, the rabbit population reaches after t equals the lag time.

05:18 Rabbit population reaches max okay it reaches the max and that time at that time when the rabbit population is max box population is y equals a over b okay and we'll label that why um why max no that'll be x max a over b and afterwards the rabbit population increases the rabbit population decreases d x over dt less than zero while the fox population continues to grow.

07:06 Okay.

07:12 And it grows at d, y, or dt is greater than zero.

07:19 So then the curve passes to region three.

07:27 And then fox population max, it matches y max.

07:38 And then the rabbit population at this point.

07:40 Point is x equals c over d.

07:50 Then right after the fox population increases, and right after that, fox population decreases, d .y over dt is less than zero...

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