At this point in the course, I lack the tools to solve differential equations. Instead, I want to present some qualitative analysis of the differential equation. I’m going to consider a restrictive type of DE.
Definition3.3.1.
A differential equation where the left-side is just \(\frac{dp}{dt}\) and the right-side is some algebraic expression in \(p\) is called an autonomous differential equation. (The independent variable \(t\) cannot appear in this expression on the left-side.)
Many natural models are described by autonomous equations, including population growth. There is a lovely piece of analysis for autonomous equations called the phase line analysis. It leads to a solid understanding of the behaviour of the model without actually having to solve the DE at all.
Subsection3.3.2Phase Line Analysis
Phase line analysis looks at the right side of an autonomous differential equation and asks for values of \(p\) which set the right side to zero. What does this mean? When the right side of the differential equation is zero, the left side is zero as well. The left side is the growth rate, so that means the growth rate is (momentarily) zero. Therefore, these values of \(p\) are values of the population where there is no growth. We call these values steady states of the population. If the population is exactly at its steady state, it will not change; steady states are constant popluations which do no grow or decline. Once I have the steady states, I investigate what happens between each steady state. Between the steady states, the right side will be either positive or negative. When it is positive, the left side is also positive, so the population has a positive growth rate and will increase. When it is negative, the left side is also negative, so the population has a negative growth rate and will decrease. This direction of growth, negative or positive, is called the trajectory of the popluation between the steady states.
Amazingly, this gives me an impressively complete understanding of the population.
If the popluation is at a steady state, it doesn’t change.
If the popluation is not at a steady state, I look at the trajectory.
If the trajectory is positive, the popluation grows approaching either the closest larger steady state or infinity.
If the trajectory is neagative, the population declines approaching either the closest smaller steady state or zero.
A phase line diagram is a visual summary of this information. It consists of a real number line, representing the function. (For populations, of course, negative values are excluded). Dots are placed on the phase line to indicate the steady states. In between the steady states, arrows are drawn to show the trajectories. Its best to see the phase line diagrams through examples.
Note that the behaviour depends on the current population, not the time. I need to know the value of the population to determine the behaviour. All the points on the phase line are values of population, not values of time.
Example3.3.2.
\begin{equation*}
\frac{dp}{dx} = p^2 - p
\end{equation*}
The right side is zero when \(p=0\) or \(p=1\text{,}\) so those are the steady states. When \(p \in (0,1)\) the right-side is negative, so the trajectory is decreasing. When \(p \in (1, \infty)\text{,}\) the right-side is positive, so the trajectory is increasing. This phase line encodes this information.
Figure3.3.3.The Phase Line Diagram for \(\frac{dp}{dx} = p^2 -
p\)
Example3.3.4.
This is an example of the logistic equation, which will be discussed in detail in Subsection 9.3.2.
The right side is zero when \(p=0\) or \(p=4\text{,}\) so those are the steady states. When \(p \in (0,4)\) the right-side is positive, so the trajectory is increasing. When \(p \in (4, \infty)\text{,}\) the right-side is negative, so the trajectory is decreasing.
Figure3.3.5.The Phase Line Diagram for \(\frac{dp}{dt} = 4p -
p^2\)
The logistic equation leads to logistic growth. Logistic growth is identified by the fact that the two trajectories point towards the steady state \(p=4\text{.}\) In logistic growth, the population wants to approach to some non-zero steady state. From below, this is growth up to some maximum and from above it is decay down to a minimum. (After exponential growth, logistic growth is the most commonly used model for populations.) Figure 3.3.6 shows both exponential and logistic growth (where the steady state for the logistic model is at \(p=6\text{.}\))
The right side factors as \(p(p-2)(p-5)\text{,}\) so it is zero when \(p=0\text{,}\)\(p=2\) or \(p=5\text{.}\) Those are the steady states. When \(p \in (0,2)\) the right-side is positive, so the trajectory is increasing. When \(p \in
(2,5)\text{,}\) the right-side is negative, so the trajectory is decreasing. When \(p \in (5,\infty)\text{,}\) the right-side is positive, so the trajectory is increasing.
Figure3.3.8.The Phase Line Diagram for \(\frac{dp}{dt} = p^3 -
7p^2 + 10p\)
