# Biological interpretation of a system of DE

In many cases, it is the adult members of the prey that are attacked mainly by the predators while the young members are better protected.

Let $$x_1$$ be the number of adult prey, $$x_2$$ the number of young prey and $$y$$ the number of predators.The following system models the dynamics of the species.

\begin{align}\frac{dx_1}{dt}&=-a_1x_1+a_2x_2-bx_1y\\ \frac{dx_2}{dt}&=nx_1-(a_1+a_2)x_2\\ \frac{dy}{dt}&=-cy+dx_1y\end{align}

Biologically interpret each of the terms of the model.

Below is my attempt. Did I interprete each term of the model correctly?

The first equation:

$$-a_1x_1$$ means that the population of adult prey will disappear by the pass of the time.

$$a_2x_2$$ means that the population of young prey have an exponential growth.

$$-bx_1y$$ means the meeting between a prey and predator, particularly $$b$$ means the success of the predator, i.e. a predator eats a prey.

2nd equation

$$nx_1$$ means that the population of adult prey have an exponential growth.

$$(-a_1+a_2)x_2$$ means that the population of young prey will disappear by the pass of the time.

3rd equation

$$-cy$$ means that the population of predators will disappear by the pass of the time.

$$dx_1y$$ means the probability that a predator being born since it eat an adult prey.

• If anyone knows where this exercise is (from a book) please tell me:) Dec 21, 2017 at 20:38

Your interpretation is basically correct, but I might phrase things somewhat differently. It also might be worthwhile to take a step back and understand how the model was built in the first place. Quoting the problem statement:

In many cases, it is the adult members of the prey that are attacked mainly by the predators while the young members are better protected.

Let $x_1$ be the number of adult prey, $x_2$ the number of young prey and $y$ the number of predators.The following system models the dynamics of the species.

\begin{align}\frac{\mathrm{d}x_1}{\mathrm{d}t}&=-a_1x_1+a_2x_2-bx_1y\\ \frac{\mathrm{d}x_2}{\mathrm{d}t}&=nx_1-(a_1+a_2)x_2\\ \frac{\mathrm{d}y}{\mathrm{d}t}&=-cy+dx_1y\end{align}

Biologically interpret each of the terms of the model.

The first sentence is telling us that we are trying to model the interaction of two species: a prey species and a predator species. It also tells us that young prey are less likely to be eaten than older prey, which means that any model of the interaction should take this into account. Mathematically, this means that we have three populations to consider: \begin{align} x_1 &= x_1(t) = \text{the number of adult prey at time $t$,} \\ x_2 &= x_2(t) = \text{the number of young prey at time $t$, and} \\ y &= y(t) = \text{the number of predators at time $t$.} \end{align} At this point, we can start building the model.

First, might ask how the population of adult prey can change. That is, what might cause $x_1$ to go up or down? Based on the statement of the problem (and common sense), there are essentially three events that might cause the population to fluctuate: (a) a young prey animal can mature into an adult prey animal, (b) an adult prey animal can be killed by a predator, and (c) an adult prey animal can leave the population through some other means (i.e. they die of old age, are eaten by a different predator, or leave the area being studied).

Taking just (a) for a moment, it is reasonable to assume that the rate at which young prey animals mature into adult prey animals is proportional to the number of young prey animals. That is, the rate at which the adult population grows due to young animals maturing is $a_2 x_2$, where $a_2$ is some constant which represents the rate at which young animals grow up. By similar reasoning, the rate at which adults die should be proportional to the population of adults, i.e. the rate should be $-a_1x_1$, where $a_1$ is a "death rate". Finally, we expect the rate of predation to be jointly proportional to the size of the prey population and the predator population (basically, more of each implies that they are more likely to encounter each other, and so more prey get eaten). This explains the $bx_1y$ term.

In plain English, the equation $$\frac{\mathrm{d}x_1}{\mathrm{d}t} = a_2x_2 - a_1x_1 - bx_1y \tag{1}$$ can be read as "The rate at which the population of adult prey changes is equal to the rate at which young animals mature into the population ($a_2x_2$), minus the rate at which adult animals die from predation ($bx_1y$) and other causes ($a_1 x_1$)."

By similar reasoning, we can parse the equation $$\frac{\mathrm{d}x_2}{\mathrm{d}t} = nx_1 - a_1x_2 - a_2x_2. \tag{2}$$ The first term represents the number of babies that are born, which is proportional to the size of the adult population; the second term is the number of young prey animals that die other causes (it is funny to me that the rate is the same for young and adult animals, but this is what the model says); and the last term is the rate at which young animals mature—note that this is exactly the rate from the previous equation. This makes sense, as we expect young animals to mature into adult animals. It is also interesting to me that this model assumes that young prey are immune from predation (note that there is no $x_2y$ term in this equation).

Finally $$\frac{\mathrm{d}y}{\mathrm{d}t} = -cy + dx_1y \tag{3}$$ says that the rate at which the predator population changes is equal to the rate at which predators die ($cy$), plus the rate at which predators are born ($dx_1y$). The model implies that this second rate is jointly proportional to the number of adult prey and the number of predators. This also makes some kind of sense, as more predators means more babies, but fewer prey means starving predators, which means fewer babies.

In summary:

1. Equation (1) models the rate at which the adult population changes.
• $a_2x_2$ represents the rate at which young prey animals mature into adult prey animals,
• $a_1x_1$ represents the rate at which adult prey animals die due to reasons other than predation, and
• $bx_1y$ represents the rate at which adult prey animals die due to predation.
2. Equation (2) models the rate at which the young prey population changes.
• $nx_1$ represents the rate at which young prey animals are born,
• $a_1x_2$ represents the rate at which young prey animals die, and
• $a_2x_2$ represents the rate at which young prey animals mature into adult prey animals (as in the previous equation).
3. Equation (3) models the rate at which the predator population changes.
• $cy$ is the rate at which predators die, and
• $dx_1 y$ is the rate at which predators are born.
• Thank you so much Xander for the clear explanation on how the model was built $\ddot\smile$. I appreciated it. Dec 28, 2017 at 20:51

I believe what you want to say instead is this:

The terms $-a_1x_1$ in the first equation and $-a_1x_2$ in the second equation are the result of the prey dying by natural causes with rate $a_1$. This is an exponential decay term.

The term $-cy$ in the third equation is also an exponential decay term for the death rate of the predators.

The terms $+a_2x_2$ in the first equation and $-a_2x_2$ in the second equation refer to the rate $a_2$ at which young prey mature into adult prey.

The term $+nx_1$ in the second equation refers to the birth rate of young prey which is proportional to the adult prey population.

Finally $-bx_1y$ is the rate at which adult prey are killed by predators, and $+dx_1y$ is the rate that the predators are able to reproduce given the available food supply (the prey). Interaction terms are typically proportional to both populations in these kind of models.

• $a_2x_2$ and $-a_2x_2$ means the same, i.e. young prey mature into adult prey? Dec 22, 2017 at 0:18
• Yes. In the first equation it says that the number of adult prey increases at the rate that the young mature into adults, and in the second equation it says that the number of young prey decreases at the rate that they mature into adults. Dec 27, 2017 at 20:53