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
266
$endgroup$
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
266
$endgroup$
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
266
$endgroup$
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
266
asked Dec 10 '18 at 12:49
TomTom
266
edited Dec 10 '18 at 13:48
Tom
asked Dec 10 '18 at 12:49
TomTom
266
asked Dec 10 '18 at 12:49
TomTom
266
asked Dec 10 '18 at 12:49
TomTom
266
266
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3033875%2fexplicit-solutions-of-sde%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3033875%2fexplicit-solutions-of-sde%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
