Explicit solutions of SDE
$begingroup$
I am trying to use random variables to represent variation in population growth models such that growth rates can be represented as random variables themselves. In the simple exponential growth case we can define the growth rate as:
$$
frac{dC}{dt} = rC
$$
where $r$ is a random variable. Integrating this over time gives:
$$
C(t) = C_0 e^{rt}
$$
which we can work out the expected value and variance (assuming that $r sim N(mu,sigma^2)$) as:
$$
E[C(t)] = C_0 e^{tmu + frac{t^2 sigma^2}{2}}\
text{Var}[C(t)] = C_0^2 e^{2tmu+sigma^2t^2}(e^{sigma^2t^2}-1)
$$
My problem is when I try to include a form of density dependence such that the growth rate is affected by the average amount of biomass in the population:
$$
frac{dC}{dt} = C(r - E[C])
$$
Can this be solved analytically? I'm not sure how to approach the problem as we cannot preform the first step to get the time dependent solution as the mean biomass of the population $E[C]$ will change over time.
calculus probability mathematical-modeling sde
asked Dec 10 '18 at 12:49
TomTom
386
$endgroup$
add a comment |
$begingroup$
I am trying to use random variables to represent variation in population growth models such that growth rates can be represented as random variables themselves. In the simple exponential growth case we can define the growth rate as:
$$
frac{dC}{dt} = rC
$$
where $r$ is a random variable. Integrating this over time gives:
$$
C(t) = C_0 e^{rt}
$$
which we can work out the expected value and variance (assuming that $r sim N(mu,sigma^2)$) as:
$$
E[C(t)] = C_0 e^{tmu + frac{t^2 sigma^2}{2}}\
text{Var}[C(t)] = C_0^2 e^{2tmu+sigma^2t^2}(e^{sigma^2t^2}-1)
$$
My problem is when I try to include a form of density dependence such that the growth rate is affected by the average amount of biomass in the population:
$$
frac{dC}{dt} = C(r - E[C])
$$
Can this be solved analytically? I'm not sure how to approach the problem as we cannot preform the first step to get the time dependent solution as the mean biomass of the population $E[C]$ will change over time.
calculus probability mathematical-modeling sde
asked Dec 10 '18 at 12:49
TomTom
386
$endgroup$
add a comment |
$begingroup$
I am trying to use random variables to represent variation in population growth models such that growth rates can be represented as random variables themselves. In the simple exponential growth case we can define the growth rate as:
$$
frac{dC}{dt} = rC
$$
where $r$ is a random variable. Integrating this over time gives:
$$
C(t) = C_0 e^{rt}
$$
which we can work out the expected value and variance (assuming that $r sim N(mu,sigma^2)$) as:
$$
E[C(t)] = C_0 e^{tmu + frac{t^2 sigma^2}{2}}\
text{Var}[C(t)] = C_0^2 e^{2tmu+sigma^2t^2}(e^{sigma^2t^2}-1)
$$
My problem is when I try to include a form of density dependence such that the growth rate is affected by the average amount of biomass in the population:
$$
frac{dC}{dt} = C(r - E[C])
$$
Can this be solved analytically? I'm not sure how to approach the problem as we cannot preform the first step to get the time dependent solution as the mean biomass of the population $E[C]$ will change over time.
calculus probability mathematical-modeling sde
asked Dec 10 '18 at 12:49
TomTom
386
$endgroup$
I am trying to use random variables to represent variation in population growth models such that growth rates can be represented as random variables themselves. In the simple exponential growth case we can define the growth rate as:
$$
frac{dC}{dt} = rC
$$
where $r$ is a random variable. Integrating this over time gives:
$$
C(t) = C_0 e^{rt}
$$
which we can work out the expected value and variance (assuming that $r sim N(mu,sigma^2)$) as:
$$
E[C(t)] = C_0 e^{tmu + frac{t^2 sigma^2}{2}}\
text{Var}[C(t)] = C_0^2 e^{2tmu+sigma^2t^2}(e^{sigma^2t^2}-1)
$$
My problem is when I try to include a form of density dependence such that the growth rate is affected by the average amount of biomass in the population:
$$
frac{dC}{dt} = C(r - E[C])
$$
Can this be solved analytically? I'm not sure how to approach the problem as we cannot preform the first step to get the time dependent solution as the mean biomass of the population $E[C]$ will change over time.
calculus probability mathematical-modeling sde
calculus probability mathematical-modeling sde
asked Dec 10 '18 at 12:49
TomTom
386
asked Dec 10 '18 at 12:49
TomTom
386
edited Dec 10 '18 at 13:48
Tom
asked Dec 10 '18 at 12:49
TomTom
386
asked Dec 10 '18 at 12:49
TomTom
386
asked Dec 10 '18 at 12:49
TomTom
386
386
add a comment |
add a comment |
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