This question is about the model of species competition. Two species that don't predate on one other yet compete for a limited food source. Their corresponding population are N1(t)andN2(t).

\begin{align}
\dot{N}_1(t) &= r_1 N_1(t) \left(1 - \frac{N_1(t)}{K_1} - b_{12} \frac{N_2}{K_1}
\right) \\
\dot{N}_2(t) &= r_2 N_2(t) \left(1 - \frac{N_2(t)}{K_2} - b_{21} \frac{N_1}{K_2}
\right)
\end{align}

Note that the term $b_{ij} \frac{N_j}{K_i}$ models inter-species competition. For example, if $ N_{2}(t) $ is large, the term negatively impacts the population growth rate of $ N_{1}(t) $.

Now that we are going to reduce the number of parameters in our model (a process known as nondimensionalization), the corresponding model is as follows:

$ \begin{align}
\frac{d u_1}{d \tau}(\tau) &= u_1(\tau) \left(1 - u_1(\tau) - a_{12}u_2(\tau)
\right) \\
\frac{d u_2}{d \tau}(\tau) &= \rho u_2(\tau) \left(1 - u_2(\tau) -
a_{21}u_1(\tau) \right)
\end{align}$

where the updated parameters are expressed using the original parameters as a reference:

$u_1 = \frac{N_1}{K_1}, \quad u_2 = \frac{N_2}{K_2}, \quad \tau = r_1 t, \quad
\rho = ?????, \ \ \ \\
a_{12} = b_{12}\frac{K_2}{K_1}, \quad a_{21} = b_{21}\frac{K_1}{K_2}$.

A): Determine $\rho$ in relation to the original equation's parameters. Explain its biological meaning (you might be able to explain it even if you couldn't figure out its formula and after running the below simulations). Hint: given an arbitrary variable $x$, determine the relationship between $\dot{x}(t) = \frac{d x}{d t}$ and $\frac{d x}{d \tau} $? For example, $t$ might be in minutes and $r_1=60$ could then mean an hour.

B): We will now run many simulations of the differential equation. In order to simulate the differential equation, write some Python code. Using the initial conditions and parameter settings listed below, plot a simulation: $u_1(0) = u_2(0) = 0.1$
$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$.

Please note: we will repeatedly change the parameters to observe how they affect the model's behaviour. There is no need to write down or plot something for this exercise; just simulate repeatedly with each of the following sets of parameter settings. This is just to give you an idea of how the model behaves when the parameters are changed.

$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 0.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 0.6$
$a_{12} = 0.6 \quad a_{21} = 0.4 \quad \rho = 5$
$a_{12} = 0.4 \quad a_{21} = 0.6 \quad \rho = 5$

C): Based on the simulation results of these parameter configurations, which appears to be more critical for the long-term survival of the species in this model: the degree to which it is suppressed by its competitor or its natural growth rate? Plot a single simulation based on the aforementioned parameter values that supports your conclusion.

D): Here, four fixed points of the differential equation must be calculated. There are three that are easier and one that is more difficult. In terms of the competition between the two species, provide a concise explanation (1/2 sentence) of what each of the four fixed points signifies. While not explicitly required for this question, the preceding simulations could serve as a beneficial reality check for the responses provided.

The general form for our system of differential equation is the following: $\dot{x}(t) = f\big(x(t) \big)$,
where $x(t)$ is the vector $[u_1(\tau), u_2(\tau)]$. Please note that $\tau$ means time with rescaled units (e.g., hour vs minutes), therefore, just think of it as time.

E): Please calculate the Jacobian Matrix $J(x) = \frac{d f}{d x}(x)$

F). The Jacobian Matrix should be evaluated at the two fixed positions that you arrive at by using the parameters:

$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 1.6$

What values of $a_{12}$ and $a_{21}$ must meet in order for these fixed points to be considered stable?

In order to achieve this, determine the conditions under which the Jacobian Matrix has the appropriate eigenvalues. It is not essential to calculate each eigenvalue separately.

Rather, pay attention to this:

> 1. The matrix's trace, or the sum of its diagonal elements, equals to the sum of eigenvalues.
> 2. The matrix's determinant is equivalent to the eigenvalues' product.

Question

07-01-2024
Mathematics

contestada

This question is about the model of species competition. Two species that don't predate on one other yet compete for a limited food source. Their corresponding population are N1(t)andN2(t).

\begin{align}
\dot{N}_1(t) &= r_1 N_1(t) \left(1 - \frac{N_1(t)}{K_1} - b_{12} \frac{N_2}{K_1}
\right) \\
\dot{N}_2(t) &= r_2 N_2(t) \left(1 - \frac{N_2(t)}{K_2} - b_{21} \frac{N_1}{K_2}
\right)
\end{align}

Note that the term $b_{ij} \frac{N_j}{K_i}$ models inter-species competition. For example, if $ N_{2}(t) $ is large, the term negatively impacts the population growth rate of $ N_{1}(t) $.

Now that we are going to reduce the number of parameters in our model (a process known as nondimensionalization), the corresponding model is as follows:

$ \begin{align}
\frac{d u_1}{d \tau}(\tau) &= u_1(\tau) \left(1 - u_1(\tau) - a_{12}u_2(\tau)
\right) \\
\frac{d u_2}{d \tau}(\tau) &= \rho u_2(\tau) \left(1 - u_2(\tau) -
a_{21}u_1(\tau) \right)
\end{align}$

where the updated parameters are expressed using the original parameters as a reference:

$u_1 = \frac{N_1}{K_1}, \quad u_2 = \frac{N_2}{K_2}, \quad \tau = r_1 t, \quad
\rho = ?????, \ \ \ \\
a_{12} = b_{12}\frac{K_2}{K_1}, \quad a_{21} = b_{21}\frac{K_1}{K_2}$.

A): Determine $\rho$ in relation to the original equation's parameters. Explain its biological meaning (you might be able to explain it even if you couldn't figure out its formula and after running the below simulations). Hint: given an arbitrary variable $x$, determine the relationship between $\dot{x}(t) = \frac{d x}{d t}$ and $\frac{d x}{d \tau} $? For example, $t$ might be in minutes and $r_1=60$ could then mean an hour.

B): We will now run many simulations of the differential equation. In order to simulate the differential equation, write some Python code. Using the initial conditions and parameter settings listed below, plot a simulation: $u_1(0) = u_2(0) = 0.1$
$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$.

Please note: we will repeatedly change the parameters to observe how they affect the model's behaviour. There is no need to write down or plot something for this exercise; just simulate repeatedly with each of the following sets of parameter settings. This is just to give you an idea of how the model behaves when the parameters are changed.

$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 0.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 0.6$
$a_{12} = 0.6 \quad a_{21} = 0.4 \quad \rho = 5$
$a_{12} = 0.4 \quad a_{21} = 0.6 \quad \rho = 5$

C): Based on the simulation results of these parameter configurations, which appears to be more critical for the long-term survival of the species in this model: the degree to which it is suppressed by its competitor or its natural growth rate? Plot a single simulation based on the aforementioned parameter values that supports your conclusion.

D): Here, four fixed points of the differential equation must be calculated. There are three that are easier and one that is more difficult. In terms of the competition between the two species, provide a concise explanation (1/2 sentence) of what each of the four fixed points signifies. While not explicitly required for this question, the preceding simulations could serve as a beneficial reality check for the responses provided.

The general form for our system of differential equation is the following: $\dot{x}(t) = f\big(x(t) \big)$,
where $x(t)$ is the vector $[u_1(\tau), u_2(\tau)]$. Please note that $\tau$ means time with rescaled units (e.g., hour vs minutes), therefore, just think of it as time.

E): Please calculate the Jacobian Matrix $J(x) = \frac{d f}{d x}(x)$

F). The Jacobian Matrix should be evaluated at the two fixed positions that you arrive at by using the parameters:

$a_{12} = 0.9 \quad a_{21} = 1.1 \quad \rho = 1.6$
$a_{12} = 1.1 \quad a_{21} = 0.9 \quad \rho = 1.6$

What values of $a_{12}$ and $a_{21}$ must meet in order for these fixed points to be considered stable?

In order to achieve this, determine the conditions under which the Jacobian Matrix has the appropriate eigenvalues. It is not essential to calculate each eigenvalue separately.

Rather, pay attention to this:

> 1. The matrix's trace, or the sum of its diagonal elements, equals to the sum of eigenvalues.
> 2. The matrix's determinant is equivalent to the eigenvalues' product.

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