for some function , then the rate of variation of is entirely dependent on the amount and not on the actual time at which that particular amount occurs. We call such a model an autonomous system, and the expression in on the right-hand side is called the rate function. Population dynamics are a typical example of autonomous systems: if a scientist is studying bacterial growth in controlled lab environment, the rate of growth doesnβt have much to do with the reading on the clock on the wall, only on the current number of bacteria.
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 first thing to notice is that at quantity levels and , we have , which means that the rate of growth is zero, and neither growth nor decay can occur. So if the quantity reaches a level of or , it will remain constant and never deviate from that level afterward.
Next, we can analyze which quantity levels will result in growth and which will result in decay. As in Section 18.1, we can determine this by solving and , but this time we will solve in terms of rather than in terms of . Using our rate equation, we can determine when the derivative is positive or negative by determining the values of that make the right-hand side of (βΆ) positive or negative. We already know that this right-hand expression is zero at and , so we just need to test around those values.
We can perform a similar analysis for concavity, but for that we need to know something about the second derivative. If we differentiate both sides of (βΆ), on the left we will be differentiating the derivative of , and so weβll get the second derivative. On the right, we could expand before differentiating so that we can just apply Pattern 12.4.3, or we can directly apply the Product Rule. But in either case we will also need to apply the Chain Rule, since we are differentiating with respect to , not .
We already know the quantity levels where the derivative is positive, negative, or zero, so we just need to combine that with an analysis of the new factor . We see that will be zero at and (since these make zero) and at , so again we test around these levels.
We see now that any solution curve that passes through the quantity level will experience a concavity change there, so we will always see inflection points at that level. However, even though the concavity also changes at above and below and also above and below , those will not become inflection points, since any solution that reaches one of those levels would become constant after that.
Itβs useful to summarize all of the information so far pictorially on a phase line. Since our behaviours are based on different ranges of -values, weβll use a vertical axis to lay out the different behaviour ranges.
Figure20.1.2.A phase line describing solution behaviours on different quantity ranges.
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 βattractionβ to or a βrepulsionβ from the constant solution lines. We often model our systems as if they are βclosed,β in the sense that no unaccounted-for outside factors can influence the systemβs behaviour. But reality is typically messier than that, and outside factors can βnudgeβ a system out of a constant, stable state into a different type of behaviour. After such a βnudgeβ, 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.
A constant function has , so, as noted in the definition, equilibrium solutions occur precisely at -levels where .
We can easily determine whether an equilibrium solution is stable, unstable, or semi-stable without going through the full analysis, but instead just considering the increasing/decreasing behaviour of solutions. A stable equilibrium must have decreasing behaviour just above it and increasing behaviour just below, and an unstable equilibrium must have the opposite. A half-stable equilibrium will have same type of behaviour both just above and just below, either both increasing or both decreasing.
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 βconcavity.β 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 βflattensβ 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 βinitial conditionβ for both solutions. In an autonomous system the behaviour is completely determined by the quantity amount and not by the independent variable . 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 werenβt actually different solutions to begin with.
This principle explains why we cannot have an inflection point at an equilibrium -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 , in Figure 20.1.3 you will find it impossible to draw a curve that passes through the equilibrium solution at level (with a horizontal slope as it does so) that meets the increasing/decreasing and concavity behaviour requirements both above and below.
Determine the equilibrium solutions by solving for . Plot these values on a phase line.
Determine ranges of increase/decrease by testing for or between the equilibrium solutions, and between equilibrium solutions and any -levels where is undefined.
Determine possible inflection levels by solving for , and also noting any -levels where is undefined. Plot these levels on the phase line.
Noting that
,
determine ranges of concavity by testing for or between equilibrium solutions, and equilibrium solutions and the levels identified in the previous step.
Add the information concerning ranges of increase/decrease and concavity to the phase line. Sketch some representative curves with various initial -levels, illustrating all of the different types of behaviours possible according to the analysis.
We have already analyzed this system several times, but we will review it once again as it will serve as a βbaseβ model that we will modify in different ways in the rest of this chapter.
for some constant of proportionality . We might use such a model when we study a population that has lots of food and space and no predators or disease. In this case, we should assume that the growth rate factor is positive, since when is positive we should observe growth, in which case should also be positive.
Letβs 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 βjumpedβ over the carrying capacity (say, by a significant number of new individuals moving or being placed into the systemβs environment), then the population would experience decline instead of growth.
To model this mathematically, we might assume that the rate of growth is still proportional to the population level, but in a proportion that varies according to how close the population level is to the carrying capacity:
The equilibrium solutions are -levels that make , and from the graph we see that this occurs at and . 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 , for we see that the rate function is positive, which translates into a positive hence an increasing . But for , the rate function is negative and so is decreasing on that range. These behaviours are in line with our intentions when we created the model. We can also identify as an unstable equilibrium as solutions above that level increase away from it, and as a stable equilibrium as solutions below increase toward it and solutions above decrease toward it.
So concavity is determined by combinations of positive/negative (where our plot in Figure 20.3.1 is above/below the horizontal axis) and positive/negative (where our plot in Figure 20.3.1 is increasing/decreasing). By symmetry of the parabola, peak in the graph occurs at .
Up to , is positive and increasing as a function in , so that is also positive. These two positives combine to a positive and a concave up .
After , the graph of starts to decrease but remains positive, so a positive combined with a negative makes a negative and a concave down .
At , the graph of becomes negative and is still decreasing, so the two negatives combine to a positive and a concave up .
Another way to modify our initial exponential-growth model is to assume that there is some βhealthyβ population threshold , below which the population is not large enough to thrive and instead dwindles away. Again we create a model where the rate of variation is proportional to the population, but this time with a variable proportion that is negative when is below the threshold and positive when is above it.
Similarly to the logistic growth model, the factor (with assumed to be positive) is the intrinsic growth rate, so that for we have approximately exponential decay.
Notice that the equilibriums have βflippedβ in nature from logistic growth. The equilibrium at is now stable, since if we have we are below the threshold and the population will wither away. However, the equilibrium at the threshold is unstable equilibrium, since below the threshold we always observe decay toward , while above the threshold the population will now thrive and grow away from the threshold.
Finally, we can combine our previous modifications into one model that takes into account both a carrying capacity, above which the environment is not able to sustain such a large population, and a threshold, below which the population is not robust enough to thrive and grow. The rate equation for this model is of the form
In this model, the equilibrium at remains stable, since it is below the threshold. The equilibrium at the threshold itself 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.
