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To perform a nullcline analysis for the given predator-prey model with an Allee effect, we need to find the nullclines for both the prey population ([tex]\(N\)[/tex]) and the predator population ([tex]\(P\)[/tex]). Nullclines are the curves where the growth rate of either the prey or the predator is zero.
Given the equations:
[tex]\[ \Delta N = r N \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a N P \][/tex]
[tex]\[ \Delta P = b N P - d P \][/tex]
let's analyze these step-by-step:
### Nullcline for [tex]\(\Delta N = 0\)[/tex]
Set [tex]\(\Delta N = 0\)[/tex]:
[tex]\[ r N \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a N P = 0 \][/tex]
This equation can be factored as:
[tex]\[ N \left[r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a P\right] = 0 \][/tex]
So, the nullcline for [tex]\(\Delta N = 0\)[/tex] occurs when either:
1. [tex]\(N = 0\)[/tex]
2. [tex]\(r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a P = 0\)[/tex]
For non-zero [tex]\(N\)[/tex]:
[tex]\[ r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P \][/tex]
Let's denote this equation as the nullcline for [tex]\(\Delta N = 0\)[/tex].
### Nullcline for [tex]\(\Delta P = 0\)[/tex]
Set [tex]\(\Delta P = 0\)[/tex]:
[tex]\[ b N P - d P = 0 \][/tex]
This equation can be factored as:
[tex]\[ P (b N - d) = 0 \][/tex]
So, the nullcline for [tex]\(\Delta P = 0\)[/tex] occurs when either:
1. [tex]\(P = 0\)[/tex]
2. [tex]\(b N - d = 0\)[/tex]
For non-zero [tex]\(P\)[/tex]:
[tex]\[ b N = d \implies N = \frac{d}{b} \][/tex]
Let's denote this equation [tex]\((N = \frac{d}{b})\)[/tex] as the nullcline for [tex]\(\Delta P = 0\)[/tex].
### Finding Equilibrium Points
To find the equilibrium points, substitute [tex]\(N = \frac{d}{b}\)[/tex] from the [tex]\(\Delta P = 0\)[/tex] nullcline into the [tex]\(\Delta N = 0\)[/tex] nullcline:
Substitute [tex]\(N = \frac{d}{b}\)[/tex] into:
[tex]\[ r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P \][/tex]
We get:
[tex]\[ r \left(\frac{\frac{d}{b}}{M} - 1\right)\left(1 - \frac{\frac{d}{b}}{K}\right) = a P \][/tex]
Simplify inside the parentheses:
[tex]\[ r \left(\frac{d}{b M} - 1\right)\left(1 - \frac{d}{bK}\right) = a P \][/tex]
This can be solved for [tex]\(P\)[/tex] for the equilibrium condition. Setting the resulting equation to zero and solving for [tex]\(P\)[/tex], we derive the equilibrium condition for [tex]\(P\)[/tex]:
[tex]\[ P = \frac{d \cdot r \left(\frac{d}{b M} - 1\right)\left(1 - \frac{d}{b K}\right)}{a b} \][/tex]
### Final Nullcline Analysis Result
Hence, the nullclines for the predator-prey model with an Allee effect are:
1. [tex]\(N = 0\)[/tex]
2. [tex]\(r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P\)[/tex]
3. [tex]\(P = 0\)[/tex]
4. [tex]\(N = \frac{d}{b}\)[/tex]
Additionally, the equilibrium point found involves:
[tex]\([P]_{{equilibrium}} = \frac{d}{a} \cdot \frac{r(-1 + \frac{d}{b M})(1 - \frac{d}{b K})}{b} \)[/tex]
This captures the necessary nullclines and the resulting equilibrium points of the system.
Given the equations:
[tex]\[ \Delta N = r N \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a N P \][/tex]
[tex]\[ \Delta P = b N P - d P \][/tex]
let's analyze these step-by-step:
### Nullcline for [tex]\(\Delta N = 0\)[/tex]
Set [tex]\(\Delta N = 0\)[/tex]:
[tex]\[ r N \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a N P = 0 \][/tex]
This equation can be factored as:
[tex]\[ N \left[r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a P\right] = 0 \][/tex]
So, the nullcline for [tex]\(\Delta N = 0\)[/tex] occurs when either:
1. [tex]\(N = 0\)[/tex]
2. [tex]\(r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) - a P = 0\)[/tex]
For non-zero [tex]\(N\)[/tex]:
[tex]\[ r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P \][/tex]
Let's denote this equation as the nullcline for [tex]\(\Delta N = 0\)[/tex].
### Nullcline for [tex]\(\Delta P = 0\)[/tex]
Set [tex]\(\Delta P = 0\)[/tex]:
[tex]\[ b N P - d P = 0 \][/tex]
This equation can be factored as:
[tex]\[ P (b N - d) = 0 \][/tex]
So, the nullcline for [tex]\(\Delta P = 0\)[/tex] occurs when either:
1. [tex]\(P = 0\)[/tex]
2. [tex]\(b N - d = 0\)[/tex]
For non-zero [tex]\(P\)[/tex]:
[tex]\[ b N = d \implies N = \frac{d}{b} \][/tex]
Let's denote this equation [tex]\((N = \frac{d}{b})\)[/tex] as the nullcline for [tex]\(\Delta P = 0\)[/tex].
### Finding Equilibrium Points
To find the equilibrium points, substitute [tex]\(N = \frac{d}{b}\)[/tex] from the [tex]\(\Delta P = 0\)[/tex] nullcline into the [tex]\(\Delta N = 0\)[/tex] nullcline:
Substitute [tex]\(N = \frac{d}{b}\)[/tex] into:
[tex]\[ r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P \][/tex]
We get:
[tex]\[ r \left(\frac{\frac{d}{b}}{M} - 1\right)\left(1 - \frac{\frac{d}{b}}{K}\right) = a P \][/tex]
Simplify inside the parentheses:
[tex]\[ r \left(\frac{d}{b M} - 1\right)\left(1 - \frac{d}{bK}\right) = a P \][/tex]
This can be solved for [tex]\(P\)[/tex] for the equilibrium condition. Setting the resulting equation to zero and solving for [tex]\(P\)[/tex], we derive the equilibrium condition for [tex]\(P\)[/tex]:
[tex]\[ P = \frac{d \cdot r \left(\frac{d}{b M} - 1\right)\left(1 - \frac{d}{b K}\right)}{a b} \][/tex]
### Final Nullcline Analysis Result
Hence, the nullclines for the predator-prey model with an Allee effect are:
1. [tex]\(N = 0\)[/tex]
2. [tex]\(r \left(\frac{N}{M} - 1\right)\left(1 - \frac{N}{K}\right) = a P\)[/tex]
3. [tex]\(P = 0\)[/tex]
4. [tex]\(N = \frac{d}{b}\)[/tex]
Additionally, the equilibrium point found involves:
[tex]\([P]_{{equilibrium}} = \frac{d}{a} \cdot \frac{r(-1 + \frac{d}{b M})(1 - \frac{d}{b K})}{b} \)[/tex]
This captures the necessary nullclines and the resulting equilibrium points of the system.
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