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Chapter 8: Problem 10
Find the equilibria of the following differential equations. $$ \frac{d N}{d t}=N \ln N, N>0 $$
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In Problems you will need to solve differential equations by separation of variables. In these problems it will not always be possible to solve explicitly for \(y\) in terms of \(t ;\) instead your solution may take the form of an implicit function relating the two variables. $$ \frac{d y}{d t}=\frac{t^{2}+1}{\cos y+\sin y} \text { where } y(0)=0 $$
You should treat \(h\) as a constant. For what values of \(h\) (if any) does each equation have equilibria? Use a graphical argument to show which of the equilibria (if any) are stable. $$ \frac{d x}{d t}=x\left(x^{2}-1\right)-h $$
In our compartment model we assumed that inflows and outflows are matched at \(q\) to keep the volume of water in the tank constant. It's often useful when modeling, for example, the flow of pollutant into a pristine environment, to consider what can occur if the inflows and outflows do not match. Let's assume that the tank initially contains a volume \(V_{0}\) of water. Water flows into the tank at rate \(q_{\mathrm{in}}\), and out of the tank at rate \(q_{\text {out. }}\) (You may assume \(q_{\text {in }}>q_{\text {out } .}\) ) Suppose that the water flowing into the tank contains a concentration \(C_{I}\) of solute. As usual we write \(C(t)\) for the concentration in the tank. (a) Show that the concentration in the tank can be modeled using a differential equation: $$ \frac{d}{d t}(C V)=q_{\text {in }} C_{I}-q_{\text {out }} C $$ (b) Previously we were able to treat \(V\) as a constant. Now \(V\) changes with time. Derive a formula for \(V(t)\). (c) By substituting your formula for \(V(t)\) into (a), derive a differential equation for \(C(t)\). (d) In general we cannot analyze the behavior of the solution \(C(t)\) using techniques from Section \(8.2 .\) Why not? (e) Let's assume \(C_{l}=0\). Then show that your equation from (c) can be written as: $$ \frac{d C}{d t}=\frac{-q_{\mathrm{in}} C}{V_{0}+\left(q_{\mathrm{in}}-q_{\mathrm{out}}\right) t} $$ (f) Assume some definite values for the constants in \((8.57):\) \(q_{\mathrm{in}}=2, q_{\text {out }}=1\), and \(V_{0}=20 .\) Assuming \(C(0)=1\), solve \((8.57)\) to find \(C(t) .\) Show that \(\lim _{t \rightarrow \infty} C(t)=0\).
Suppose that a fish population evolves according to a logistic equation and
that fish are harvested at a rate proportional to the population size. If
\(N(t)\) denotes the population size at time \(t\), then
$$
\frac{d N}{d t}=r N\left(1-\frac{N}{K}\right)-h N
$$
Assume that \(r=2\) and \(K=1000\).
(a) Find possible equilibria and use the graphical approach to discuss their
stability, when \(h=0.1\).
(b) Show that if \(h
A reversible chemical reaction between chemicals \(A\) and \(B\) produces a product C: \(A+B \rightleftharpoons\). We modeled this reaction in Section 8.3.3 using a differential equation for the amount of \(C\) produced: $$ \frac{d x}{d t}=k_{A B}(a-x)(b-x)-k_{C} x $$ Here \(x(t)\) is the amount of \(C\) at time \(t, a\) is the initial amount of chemical \(A, b\) is the initial amount of \(B\), and \(k_{A B}\) and \(k_{C}\) are respectively the rate constants for the reaction that creates \(C\) and for the decay of \(C\) back into \(A\) and \(B\). Explain what each term in (8.63) represents and how the equation is derived.
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