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Consider the following system and analyze its behavior.

$$\begin{array}{rl} \frac{dA}{dt} &= A \left( 2-\frac{A}{5000}-\frac{L}{100} \right)\\ \frac{dL}{dt} &= L \left(-\frac{1}{2}+\frac{A}{10000} \right)\end{array}$$

The analysis

It has $3$ equilibrium points. I know the stability of the three points but I am not sure if I interpret the meaning of them correctly according to the phase portrait. $x_1, x_2$ are saddle points and $x_3$ is stable point they are

$$\overline x_1=(0,0)$$

$$\overline x_2=(10000,0)$$

$$\overline x=(5000,100)$$

According to the phase portrait I think the behavior of the system is described as:

For every point $(A,L)$ given in the first quadrant, where $A$ is the number of aphids at time $t$ and $L$ is the number of lady-bugs at time $t$, we'll have that in the future, the maximum number of lady-bugs and aphids it's going to be $(5000,100)$ respectively. This also means that both population will never going to extinct.

Even in the case where there were just a few number (near 0) of lady-bugs, their population will grow and will be establish also.

My question

Did I miss something important in the biological description to the phase portrait?

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user441848
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    You seem to once put bars over your equilibrum conditions and once not. I am not sure they are exactly the same. Also, I think you once meant $\bar x_3$ instead $\bar x$. Note that the usual is to use a hat rather than a bar for an equlibrium. – Remi.b Jan 29 '18 at 19:35

1 Answers1

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I don't think you missed anything important!

You could investigate the cyclic behaviour around the equilibrium. For example, looking at the variable aphid population size can start above its equilibrium point, then overshoot it and overshoot it again to finally reach the equilibrium point. If you zoom close to the equilibrium point, you might see long cyclic behaviour before ever reaching the equilibrium.

You sure could do all kind of further analyses such as a stability analysis or searching for cyclical equilibrium but it is easy to see with this simple model that nothing very complicated will arise. You could also investigate for what parameter values this classic pattern breaks up if it does.

Another option is to make the model more complex by introducing a carrying capacity (and then investigate under what aphid growth rate the behaviour becomes chaotic), or population structure, or whatever.

You might learn much from it but you might still want to have a look at the post What prevents predator overpopulation?.

I highly recommend the book A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Otto and Troy. It will show the typical kind of analyses that can be done for this kind of systems.

Mithoron
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Remi.b
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  • I don't see a cyclic behaviour. Cyclic means something circular ? – user441848 Jan 29 '18 at 20:37
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    If you start at (15000, 300), you'll circle around the equilibrium. Maybe not a full circle though. I don't know. But both variables are going to overshoot the equilibrium at least twice. – Remi.b Jan 29 '18 at 20:58
  • You're right, haven't notice. Also you mention that I could make the model more complex by introducing a carrying capacity. Exactly how would be that term? I mean something like +1/k or -1/k – user441848 Jan 29 '18 at 21:12
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    @Anneliset. Remember, when you are modeling biological data like this you are plotting means, but in the real world you are going to have some added noise. That means that even if you have a stable fixed point like in this phase diagram, if you actually simulate data rather than means you are going to oscillate a bit around it because a noise term keeps you in the 'cycle' and prevents settling to perfect stability. – Bryan Krause Jan 29 '18 at 21:28
  • Carrying capacity would typically like like an added $\frac{K-N}{K}$ term that multiply the rest. See logistic population growth. It is quite classic. You might want to read an intro book on the subject rather than having to reinvent the wheel by yourself. You might want to have a look at a population biology books. I had one in mind which is apparently very good but I can't remember the title or author right now. – Remi.b Jan 29 '18 at 21:34
  • You can ask for book recommendation on another post if you want. Here is a book recommendation post for population genetics. – Remi.b Jan 29 '18 at 21:34
  • I was wondering, time $t$ is measured in what units? days or months? – user441848 Jan 30 '18 at 02:05
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    Typically measured in generation time. Your model must assume equal generation time (and non-overlapping generations) or must consider that the values are adjusted for differential generation time. – Remi.b Jan 30 '18 at 04:51
  • Oh ok, I was observing the model again and I've seen that there exist competence between aphids due this term $-\frac{A^2}{5000}$, why aphids compete in nature? – user441848 Jan 30 '18 at 23:45
  • Sure.... there is always competition among individuals in pretty much any species. – Remi.b Jan 30 '18 at 23:56
  • But you are a biologist , you must know why they compete.Is it for food or ...? (I won't bother again, this is my last question:) So if you don't mind, answer me. – user441848 Jan 30 '18 at 23:59
  • Well... I am not an aphid biologist! I am a theoretician (mainly numerical stuff, not much analytical) in population genetics. Also, it is the kind of question that would deserve its own post. I don't know the answer and I typically don't seek much in the literature for a follow-up question. I (or other users) might do it on another post. There is probably competition over habitat (incl. for leave where they make their galls), food, protection from ants, etc... – Remi.b Jan 31 '18 at 00:18
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    You can just go on Google Scholar > Aphid competition and you'll find some answers. Good luck @Annelise! – Remi.b Jan 31 '18 at 00:18
  • @Remi.b The model already has a logistic/carrying capacity component; the second term of the aphid equation can be rewritten as $-A^2/5000$, which could be interpreted as a carrying capacity of 5000 (or that $r/K = 1/5000$). – fileunderwater Mar 02 '18 at 08:52