13.1 State-space diagrams

Differential equation. Given the differential equation and initial condition

where $a>0$ is a constant, the value of the function $y(t)$ would

There's a hole in the bucket. Water flows into a bucket at constant rate $I$. There is a hole in the container. Explain the model

Analyze the behaviour predicted. What would the height be after a long time? Is this result always valid, or is an additional assumption needed? (hint: recall Example 11.2.3 in the text.)

Cubical crystal. A crystal grows inside a medium in a cubical shape with side length $x$ and volume $V.$ The rate of change of the volume is given by

where $k$ and $V_{0}$ are positive constants.

Logistic growth with proportional harvesting. Consider a fish population of density $N(t)$ growing at rate $g(N)$, with harvesting, so that the population satisfies the differential equation

Now assume that the growth rate is logistic, so $g(N)= r N \frac{(K - N)}{K}$ where $r, K>0$ are constant. Assume that the rate of harvesting is proportional to the population size, so that

where $E$, the effort of the fishermen, and $q$, the catchability of this type of fish, are positive constants.

Use qualitative methods discussed in this chapter to analyze the behaviour of this equation. Under what conditions does this lead to a sustainable fishery?

Logistic growth with constant number harvesting. Consider the same fish population as in Exercise 4, but this time assume that the rate of harvesting is fixed, regardless of the population
size, so that

where $H$ is a constant number of fish being caught and removed per unit time. Analyze this revised model and compare it to the previous results.