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Function for Differential Equation for Malthusian Population Approximation
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I just rediscovered a program I made a few years ago on Khan Academy to model the Malthusian population curve, based on a differential equation, where $K$ is the population limit, $N$ is the population, and $t$ is time.
$$fracdNdt=frac18logt+frac(K-N)1.69fracsin(t)t$$ It's sort of like a logistic curve, but with some tweaks to better fit real-life.
I was wondering how I should go about finding a function for $y$, the population, with respect to $x$, time. I already know the general logistic growth equation $$fracdNdt=rt(1-fracNK)$$ and how to get to it, but I'm not sure how to tackle my differential equation.
Thanks in advance.
The numerical constants in the first equation are mostly just arbitrary - feel free to replace them with simpler integers if you like.
differential-equations
asked Jul 18 at 23:51
RayDansh
882214
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up vote
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I just rediscovered a program I made a few years ago on Khan Academy to model the Malthusian population curve, based on a differential equation, where $K$ is the population limit, $N$ is the population, and $t$ is time.
$$fracdNdt=frac18logt+frac(K-N)1.69fracsin(t)t$$ It's sort of like a logistic curve, but with some tweaks to better fit real-life.
I was wondering how I should go about finding a function for $y$, the population, with respect to $x$, time. I already know the general logistic growth equation $$fracdNdt=rt(1-fracNK)$$ and how to get to it, but I'm not sure how to tackle my differential equation.
Thanks in advance.
The numerical constants in the first equation are mostly just arbitrary - feel free to replace them with simpler integers if you like.
differential-equations
asked Jul 18 at 23:51
RayDansh
882214
1
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I just rediscovered a program I made a few years ago on Khan Academy to model the Malthusian population curve, based on a differential equation, where $K$ is the population limit, $N$ is the population, and $t$ is time.
$$fracdNdt=frac18logt+frac(K-N)1.69fracsin(t)t$$ It's sort of like a logistic curve, but with some tweaks to better fit real-life.
I was wondering how I should go about finding a function for $y$, the population, with respect to $x$, time. I already know the general logistic growth equation $$fracdNdt=rt(1-fracNK)$$ and how to get to it, but I'm not sure how to tackle my differential equation.
Thanks in advance.
The numerical constants in the first equation are mostly just arbitrary - feel free to replace them with simpler integers if you like.
differential-equations
asked Jul 18 at 23:51
RayDansh
882214
I just rediscovered a program I made a few years ago on Khan Academy to model the Malthusian population curve, based on a differential equation, where $K$ is the population limit, $N$ is the population, and $t$ is time.
$$fracdNdt=frac18logt+frac(K-N)1.69fracsin(t)t$$ It's sort of like a logistic curve, but with some tweaks to better fit real-life.
I was wondering how I should go about finding a function for $y$, the population, with respect to $x$, time. I already know the general logistic growth equation $$fracdNdt=rt(1-fracNK)$$ and how to get to it, but I'm not sure how to tackle my differential equation.
Thanks in advance.
The numerical constants in the first equation are mostly just arbitrary - feel free to replace them with simpler integers if you like.
differential-equations
asked Jul 18 at 23:51
RayDansh
882214
edited Jul 19 at 17:20
asked Jul 18 at 23:51
RayDansh
882214
asked Jul 18 at 23:51
RayDansh
882214
asked Jul 18 at 23:51
RayDansh
882214
882214
1
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39
add a comment |Â
1
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39
1
1
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39
add a comment |Â
1 Answer
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Let $N_K = N-K$, then we have a slightly simpler equation
$$ fracdN_Kdt + fracasin ttN_K = fracbln t $$
Using the standard first-order method, an integrating factor can be found as
$$ mu(a,t) = expleft(aint_0^t fracsin tautau dtauright) = expbig(a operatornameSi(t) big) $$
where $operatornameSi(t)$ is the sine integral. Then
$$ mu fracdN_Kdt + fracdmudtN_K = fracddtbig(mu N_Kbig) = fracbmuln t $$
$$ implies N_K = frac1muint fracbmuln t dt = be^-aoperatornameSi(t)left[int_0^t frace^aoperatornameSi(tau)ln tau dtau + cright] $$
If $N(0) = N_0$ then we have the solution
$$ N(t) = K + (N_0-K)e^-aoperatornameSi(t) + be^-aoperatornameSi(t)int_0^t frace^aoperatornameSi(tau)ln tau dtau $$
The singularity at $t=1$ can be resolved by the Cauchy principal value. You may notice that this population model grows without bounds, similar to the logarithmic integral.
answered Jul 19 at 17:00
Dylan
11.4k31026
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Let $N_K = N-K$, then we have a slightly simpler equation
$$ fracdN_Kdt + fracasin ttN_K = fracbln t $$
Using the standard first-order method, an integrating factor can be found as
$$ mu(a,t) = expleft(aint_0^t fracsin tautau dtauright) = expbig(a operatornameSi(t) big) $$
where $operatornameSi(t)$ is the sine integral. Then
$$ mu fracdN_Kdt + fracdmudtN_K = fracddtbig(mu N_Kbig) = fracbmuln t $$
$$ implies N_K = frac1muint fracbmuln t dt = be^-aoperatornameSi(t)left[int_0^t frace^aoperatornameSi(tau)ln tau dtau + cright] $$
If $N(0) = N_0$ then we have the solution
$$ N(t) = K + (N_0-K)e^-aoperatornameSi(t) + be^-aoperatornameSi(t)int_0^t frace^aoperatornameSi(tau)ln tau dtau $$
The singularity at $t=1$ can be resolved by the Cauchy principal value. You may notice that this population model grows without bounds, similar to the logarithmic integral.
answered Jul 19 at 17:00
Dylan
11.4k31026
add a comment |Â
up vote
1
down vote
accepted
Let $N_K = N-K$, then we have a slightly simpler equation
$$ fracdN_Kdt + fracasin ttN_K = fracbln t $$
Using the standard first-order method, an integrating factor can be found as
$$ mu(a,t) = expleft(aint_0^t fracsin tautau dtauright) = expbig(a operatornameSi(t) big) $$
where $operatornameSi(t)$ is the sine integral. Then
$$ mu fracdN_Kdt + fracdmudtN_K = fracddtbig(mu N_Kbig) = fracbmuln t $$
$$ implies N_K = frac1muint fracbmuln t dt = be^-aoperatornameSi(t)left[int_0^t frace^aoperatornameSi(tau)ln tau dtau + cright] $$
If $N(0) = N_0$ then we have the solution
$$ N(t) = K + (N_0-K)e^-aoperatornameSi(t) + be^-aoperatornameSi(t)int_0^t frace^aoperatornameSi(tau)ln tau dtau $$
The singularity at $t=1$ can be resolved by the Cauchy principal value. You may notice that this population model grows without bounds, similar to the logarithmic integral.
answered Jul 19 at 17:00
Dylan
11.4k31026
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $N_K = N-K$, then we have a slightly simpler equation
$$ fracdN_Kdt + fracasin ttN_K = fracbln t $$
Using the standard first-order method, an integrating factor can be found as
$$ mu(a,t) = expleft(aint_0^t fracsin tautau dtauright) = expbig(a operatornameSi(t) big) $$
where $operatornameSi(t)$ is the sine integral. Then
$$ mu fracdN_Kdt + fracdmudtN_K = fracddtbig(mu N_Kbig) = fracbmuln t $$
$$ implies N_K = frac1muint fracbmuln t dt = be^-aoperatornameSi(t)left[int_0^t frace^aoperatornameSi(tau)ln tau dtau + cright] $$
If $N(0) = N_0$ then we have the solution
$$ N(t) = K + (N_0-K)e^-aoperatornameSi(t) + be^-aoperatornameSi(t)int_0^t frace^aoperatornameSi(tau)ln tau dtau $$
The singularity at $t=1$ can be resolved by the Cauchy principal value. You may notice that this population model grows without bounds, similar to the logarithmic integral.
answered Jul 19 at 17:00
Dylan
11.4k31026
Let $N_K = N-K$, then we have a slightly simpler equation
$$ fracdN_Kdt + fracasin ttN_K = fracbln t $$
Using the standard first-order method, an integrating factor can be found as
$$ mu(a,t) = expleft(aint_0^t fracsin tautau dtauright) = expbig(a operatornameSi(t) big) $$
where $operatornameSi(t)$ is the sine integral. Then
$$ mu fracdN_Kdt + fracdmudtN_K = fracddtbig(mu N_Kbig) = fracbmuln t $$
$$ implies N_K = frac1muint fracbmuln t dt = be^-aoperatornameSi(t)left[int_0^t frace^aoperatornameSi(tau)ln tau dtau + cright] $$
If $N(0) = N_0$ then we have the solution
$$ N(t) = K + (N_0-K)e^-aoperatornameSi(t) + be^-aoperatornameSi(t)int_0^t frace^aoperatornameSi(tau)ln tau dtau $$
The singularity at $t=1$ can be resolved by the Cauchy principal value. You may notice that this population model grows without bounds, similar to the logarithmic integral.
answered Jul 19 at 17:00
Dylan
11.4k31026
edited Jul 20 at 14:28
answered Jul 19 at 17:00
Dylan
11.4k31026
answered Jul 19 at 17:00
Dylan
11.4k31026
answered Jul 19 at 17:00
Dylan
11.4k31026
11.4k31026
add a comment |Â
add a comment |Â
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1
What you call ''equations'' are not equations because there is no $=$ in them. What is $y$ in $frac18log(x)+frac(K-N_o)1.69fracsin(x)x$ ? Sorry I cannot understand your question and what you are looking for.
– JJacquelin
Jul 19 at 5:18
@JJacquelin Sorry for the confusion. I've edited it now.
– RayDansh
Jul 19 at 14:39