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How is this population model different from the Logistic model?
Logistic population model questionMathematicals biologyLogistic differential equation to model populationUsing Phase planes, how do I find graphically the equilibria and their stability of a logistic growth model??Population Growth problem using Malthusian Law and the Logistic ModelIn what year does the population reach $75%$ capacity?Unknown population modelExponential Differential Equation of Population Growth ModelWhy the Logistic Differential Equation Accurately Models PopulationPopulation growth model with fishing term (logistic differential equation)
$begingroup$
I have this following population model: $$ fracdXdt = f(X) = pX(1-fracXY)(fracXZ -1) $$
and I have to compare it to the following Logistic Growth model: $$ fracdXdt = f(X) = pX(1-fracXY)$$ where p = growth rate and Y is the carrying capacity of the population?
What does the term $(fracXZ -1)$ do in order to differ it from the standard Logistic model and what kind of populations could I model with this altered version?
calculus ordinary-differential-equations derivatives
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
$endgroup$
add a comment |
$begingroup$
I have this following population model: $$ fracdXdt = f(X) = pX(1-fracXY)(fracXZ -1) $$
and I have to compare it to the following Logistic Growth model: $$ fracdXdt = f(X) = pX(1-fracXY)$$ where p = growth rate and Y is the carrying capacity of the population?
What does the term $(fracXZ -1)$ do in order to differ it from the standard Logistic model and what kind of populations could I model with this altered version?
calculus ordinary-differential-equations derivatives
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
$endgroup$
add a comment |
$begingroup$
I have this following population model: $$ fracdXdt = f(X) = pX(1-fracXY)(fracXZ -1) $$
and I have to compare it to the following Logistic Growth model: $$ fracdXdt = f(X) = pX(1-fracXY)$$ where p = growth rate and Y is the carrying capacity of the population?
What does the term $(fracXZ -1)$ do in order to differ it from the standard Logistic model and what kind of populations could I model with this altered version?
calculus ordinary-differential-equations derivatives
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
$endgroup$
I have this following population model: $$ fracdXdt = f(X) = pX(1-fracXY)(fracXZ -1) $$
and I have to compare it to the following Logistic Growth model: $$ fracdXdt = f(X) = pX(1-fracXY)$$ where p = growth rate and Y is the carrying capacity of the population?
What does the term $(fracXZ -1)$ do in order to differ it from the standard Logistic model and what kind of populations could I model with this altered version?
calculus ordinary-differential-equations derivatives
calculus ordinary-differential-equations derivatives
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
asked Mar 29 at 16:21
glaceauoxfordglaceauoxford
61
61
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
For small $X$, your model with an $fracXZ-1$ factor, hereafter the $Z$-model, obtained $dotXapprox -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $dotXapprox pX$.
If $X$ were much larger than $Y$, on the logistic model $dotXapprox-fracpX^2Y$ would lead to population decay, which is also true of the $Z$ model's approximation $dotXapprox-fracpX^3YZ$. These asymptotic results are somewhat different though, in that the former implies $frac1X$ grows approximately linearly, while the latter has this behaviour for $frac1X^2$ instead, so that the decay is slower.
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
$endgroup$
add a comment |
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$begingroup$
For small $X$, your model with an $fracXZ-1$ factor, hereafter the $Z$-model, obtained $dotXapprox -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $dotXapprox pX$.
If $X$ were much larger than $Y$, on the logistic model $dotXapprox-fracpX^2Y$ would lead to population decay, which is also true of the $Z$ model's approximation $dotXapprox-fracpX^3YZ$. These asymptotic results are somewhat different though, in that the former implies $frac1X$ grows approximately linearly, while the latter has this behaviour for $frac1X^2$ instead, so that the decay is slower.
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
$endgroup$
add a comment |
$begingroup$
For small $X$, your model with an $fracXZ-1$ factor, hereafter the $Z$-model, obtained $dotXapprox -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $dotXapprox pX$.
If $X$ were much larger than $Y$, on the logistic model $dotXapprox-fracpX^2Y$ would lead to population decay, which is also true of the $Z$ model's approximation $dotXapprox-fracpX^3YZ$. These asymptotic results are somewhat different though, in that the former implies $frac1X$ grows approximately linearly, while the latter has this behaviour for $frac1X^2$ instead, so that the decay is slower.
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
$endgroup$
add a comment |
$begingroup$
For small $X$, your model with an $fracXZ-1$ factor, hereafter the $Z$-model, obtained $dotXapprox -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $dotXapprox pX$.
If $X$ were much larger than $Y$, on the logistic model $dotXapprox-fracpX^2Y$ would lead to population decay, which is also true of the $Z$ model's approximation $dotXapprox-fracpX^3YZ$. These asymptotic results are somewhat different though, in that the former implies $frac1X$ grows approximately linearly, while the latter has this behaviour for $frac1X^2$ instead, so that the decay is slower.
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
$endgroup$
For small $X$, your model with an $fracXZ-1$ factor, hereafter the $Z$-model, obtained $dotXapprox -pX$ so small populations exponentially decay into extinction, whereas in the logistic model they exponentially grow until they're no longer small because $dotXapprox pX$.
If $X$ were much larger than $Y$, on the logistic model $dotXapprox-fracpX^2Y$ would lead to population decay, which is also true of the $Z$ model's approximation $dotXapprox-fracpX^3YZ$. These asymptotic results are somewhat different though, in that the former implies $frac1X$ grows approximately linearly, while the latter has this behaviour for $frac1X^2$ instead, so that the decay is slower.
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
answered Mar 29 at 23:15
J.G.J.G.
32.8k23250
32.8k23250
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