?A phase trajectory is shown for populations of rabbits (R) and foxes (F). (a) Describe

1. For each predator-prey system, determine which of the variables, or , represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the predators or by other factors as well? Do the predators feed only on the prey or do they have additional food sources? Explain. (a) (b)

Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance). Decide whether each system describes competition or cooperation and explain why it is a reasonable model. (Ask yourself what effect an increase in one species has on the growth rate of the other.) (a) (b)

The system of differential equations \(\frac{d x}{d t}=0.5 x-0.004 x^{2}-0.001 x y\) \(\frac{d y}{d t}=0.4 y-0.001 y^{2}-0.002 x y\) is a model for the populations of two species. (a) Does the model describe cooperation, or competition, or a predator-prey relationship? (b) Find the equilibrium solutions and explain their significance.

Lynx eat snowshoe hares and snowshoe hares eat woody plants like willows. Suppose that, in the absence of hares, the willow population will grow exponentially and the lynx population will decay exponentially. In the absence of lynx and willow, the hare population will decay exponentially. If , and represent the populations of these three species at time , write a system of differential equations as a model for their dynamics. If the constants in your equation are all positive, explain why you have used plus or minus signs.

A phase trajectory is shown for populations of rabbits (R) and foxes (F). (a) Describe how each population changes as time goes by. (b) Use your description to make a rough sketch of the graphs of R and F as functions of time.

A phase trajectory is shown for populations of rabbits (R) and foxes (F). (a) Describe how each population changes as time goes by. (b) Use your description to make a rough sketch of the graphs of R and F as functions of time.

Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory.

Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory.

In Example 1(b) we showed that the rabbit and wolf populations satisfy the differential equation (a) By solving this separable differential equation, show that where C is a constant. (b) It is impossible to solve this equation for W as an explicit function of R (or vice versa). Use a computer to graph the implicitly defined solution curve that passes through the point s1000, 40d and compare with Figure 3.

Populations of aphids and ladybugs are modeled by the equations (a) Find the equilibrium solutions and explain their significance. (b) Find an expression for dL/dA. (c) The direction field for the differential equation in part (b) is shown. Use it to sketch a phase portrait. What do the phase trajectories have in common? (d) Suppose that at time t = 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. (e) Use part (d) to make rough sketches of the aphid and ladybug populations as functions of t. How are the graphs related to each other?

In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let’s modify those equations as follows: (a) According to these equations, what happens to the rabbit population in the absence of wolves? (b) Find all the equilibrium solutions and explain their significance. (c) The figure shows the phase trajectory that starts at the point s1000, 40d. Describe what eventually happens to the rabbit and wolf populations. (d) Sketch graphs of the rabbit and wolf populations as functions of time.

In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: (a) In the absence of ladybugs, what does the model predictabout the aphids? (b) Find the equilibrium solutions. (c) Find an expression for dL/dA. (d) Use a computer to draw a direction field for the differential equation in part (c). Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common? (e) Suppose that at time t ? 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. (f) Use part (e) to make rough sketches of the aphid and ladybug populations as functions of t. How are the graphs related to each other?