Calculus Concepts and Modelling: An introduction to the concepts of calculus

We can determine a lot about the general behavioural characteristics of solutions to autonomous systems without actually solving the rate equation, by applying the concepts we learned in Chapter 18.

The constant solutions are an important feature of our analysis in Example 20.1.1, as they can be boundaries between different types of solution behaviours, both for increasing/decreasing and concavity. But also notice that solution curves near the constant solutions tend to exhibit either an “attraction” to or a “repulsion” from the constant solution lines. We often model our systems as if they are “closed,” in the sense that no unaccounted-for outside factors can influence the system’s behaviour. But reality is typically messier than that, and outside factors can “nudge” a system out of a constant, stable state into a different type of behaviour. After such a “nudge”, sometimes the system gently trends back to that original constant state, and sometimes the system reacts violently and moves quickly away from the original constant state.

So, for example, the system in Example 20.1.1 has a stable equilibrium \(q(t) = -1\) and an unstable equilibrium \(q(t) = 2\text{.}\)

A useful analogy for stable/unstable behaviour is to think of taking a bicycle ride along a path. The equilibrium state is to steer straight along the center of the path. If the sides of the path are curved upward, as if in a valley with steep sides, then any deviation from the center of the path will cause the bicycle to tend back towards that central equilibrium. On the other hand, if the sides of the path are curved downward, as if the path runs along the top of a berm, then any deviation from the center of the path will cause the rider to go careening down a slope away from the path. In a way, we could imagine that all of the solution curves stitch together to create a surface, and there is a third dimension, perpendicular to the plane where we have sketched our representative solutions, for another type of “concavity.” Near a stable equilibrium, this solution-curve surface is concave up (out of the page), with the equilibrium running along the floor of a depression. Near an unstable equilibrium, the solution-curve surface is concave down (into the page), with the equilibrium running along the peak of a long hill. Near a semi-stable equilibrium, the solution-curve surface “flattens” out between two regions of the same concavity, like a narrow horizontal shelf on the side of a hill.

We further remark on why the equilibrium solutions in Example 20.1.1 are not considered inflection points, even though the concavity of solutions changes from below to above each of those equilibriums. We take it as a general assumption that two different solution curves cannot intersect, since if they did then that point of intersection would act like an “initial condition” for both solutions. In an autonomous system the behaviour is completely determined by the quantity amount \(q\) and not by the independent variable \(t\text{.}\) So if two solution curves met at the same level, they would exhibit the same behaviour from the point forward, and, extrapolating backward, they should have exhibited the same behaviour to get to that point. Which means they weren’t actually different solutions to begin with.

This principle explains why we cannot have an inflection point at an equilibrium \(q\)-level: because solution curves cannot actually cross over an equilibrium solution, and so an individual solution curve cannot change its concavity there. Indeed, noting that equilibrium solutions necessarily have \(q'(t) = 0\text{,}\) in Figure 20.1.3 you will find it impossible to draw a curve that passes through the equilibrium solution at level \(q = -1\) (with a horizontal slope as it does so) that meets the increasing/decreasing and concavity behaviour requirements both above and below.

Finally, here is a procedure we can use to analyze solution behaviour for an autonomous system without actually solving it.

We have already analyzed this system several times, but we will review it once again as it will serve as a “base” model that we will modify in different ways in the rest of this chapter.

Let’s consider a modification to our initial population model. Suppose as the population grows, competition for space and food increases and the rate of growth slows. We might identify a particular carrying capacity of the environment: where if the population starts to approach the carrying capacity, the growth rate is significantly reduced, and if the population somehow “jumped” over the carrying capacity (say, by a significant number of new individuals moving or being placed into the system’s environment), then the population would experience decline instead of growth.

The equilibrium solutions are \(q\)-levels that make \(f(q) = 0\text{,}\) and from the graph we see that this occurs at \(q = 0\) and \(q = M\text{.}\) These make sense, since if the population cannot grow if there are no individuals at all, and also if the population is at the carrying capacity it will not be able to grow, but it also will not decline as the system tendency will be to try to grow but will be impeded by the space/food limits.

Assuming \(q \gt 0\text{,}\) for \(0 \lt q \lt M\) we see that the rate function is positive, which translates into a positive \(dq/dt\) hence an increasing \(q\text{.}\) But for \(q \gt M\text{,}\) the rate function is negative and so \(q\) is decreasing on that range. These behaviours are in line with our intentions when we created the model. We can also identify \(q = 0\) as an unstable equilibrium as solutions above that level increase away from it, and \(q = M\) as a stable equilibrium as solutions below increase toward it and solutions above decrease toward it.

We now have enough information to draw a phase line and sketch some representative solution curves.

Notice that the equilibriums have “flipped” in nature from logistic growth. The equilibrium at \(q(t) = 0\) is now stable, since if we have \(q \approx 0^+\) we are below the threshold and the population will wither away. However, the equilibrium \(q(t) = T\) at the threshold is unstable equilibrium, since below the threshold we always observe decay toward \(0\text{,}\) while above the threshold the population will now thrive and grow away from the threshold.

In this model, the equilibrium at \(q(t) = 0\) remains stable, since it is below the threshold. The equilibrium at the threshold itself \(q(t) = T\) is again unstable, as populations below it will decay and those above it will thrive. The equilibrium at the carrying capacity is stable, since populations approaching that level will significantly slow in their growth, and populations above that level will decay as the environment cannot supprt them.