Fixed points of a differential equation
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My assignment is to determine fixed points of the differential equation
$$frac{dN}{dt} = left(frac{aN}{(1+N)} - b -cNright)N$$ where $a,b,c > 0$ and find out their stability.
I do understand that concerning differential equations, a fixed point is defined as the $N$ which solves the equation $N = f(N) cdot N$. For some reason I cant even start since I only get rubbish when trying to solve for $N$ from the equation. Any suggestions where to start?
ordinary-differential-equations fixedpoints
asked Jan 23 at 16:16
OliverOliver
12
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add a comment |
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My assignment is to determine fixed points of the differential equation
$$frac{dN}{dt} = left(frac{aN}{(1+N)} - b -cNright)N$$ where $a,b,c > 0$ and find out their stability.
I do understand that concerning differential equations, a fixed point is defined as the $N$ which solves the equation $N = f(N) cdot N$. For some reason I cant even start since I only get rubbish when trying to solve for $N$ from the equation. Any suggestions where to start?
ordinary-differential-equations fixedpoints
asked Jan 23 at 16:16
OliverOliver
12
$endgroup$
$begingroup$
I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
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– jnez71
Jan 23 at 18:25
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I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
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– Oliver
Jan 23 at 18:43
add a comment |
$begingroup$
My assignment is to determine fixed points of the differential equation
$$frac{dN}{dt} = left(frac{aN}{(1+N)} - b -cNright)N$$ where $a,b,c > 0$ and find out their stability.
I do understand that concerning differential equations, a fixed point is defined as the $N$ which solves the equation $N = f(N) cdot N$. For some reason I cant even start since I only get rubbish when trying to solve for $N$ from the equation. Any suggestions where to start?
ordinary-differential-equations fixedpoints
asked Jan 23 at 16:16
OliverOliver
12
$endgroup$
My assignment is to determine fixed points of the differential equation
$$frac{dN}{dt} = left(frac{aN}{(1+N)} - b -cNright)N$$ where $a,b,c > 0$ and find out their stability.
I do understand that concerning differential equations, a fixed point is defined as the $N$ which solves the equation $N = f(N) cdot N$. For some reason I cant even start since I only get rubbish when trying to solve for $N$ from the equation. Any suggestions where to start?
ordinary-differential-equations fixedpoints
ordinary-differential-equations fixedpoints
asked Jan 23 at 16:16
OliverOliver
12
asked Jan 23 at 16:16
OliverOliver
12
edited Jan 23 at 18:05
Oliver
asked Jan 23 at 16:16
OliverOliver
12
asked Jan 23 at 16:16
OliverOliver
12
asked Jan 23 at 16:16
OliverOliver
12
12
$begingroup$
I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
$endgroup$
– jnez71
Jan 23 at 18:25
$begingroup$
I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
$endgroup$
– Oliver
Jan 23 at 18:43
add a comment |
$begingroup$
I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
$endgroup$
– jnez71
Jan 23 at 18:25
$begingroup$
I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
$endgroup$
– Oliver
Jan 23 at 18:43
$begingroup$
I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
$endgroup$
– jnez71
Jan 23 at 18:25
$begingroup$
I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
$endgroup$
– jnez71
Jan 23 at 18:25
$begingroup$
I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
$endgroup$
– Oliver
Jan 23 at 18:43
$begingroup$
I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
$endgroup$
– Oliver
Jan 23 at 18:43
add a comment |
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I'm not sure about your definition of fixed-point. I would consider the fixed-points to be the values $N^*$ that solve $Phi(N^*,t) = N^* forall t$, where $Phi$ is the flow-map of the ODE. And note that the fixed-points will coincide with the equilibrium-points, i.e. if your ODE is $frac{dN}{dt}=f(N)$ then we have $f(N^*)=0$. In your case, $f(N^*)=0$ is just a cubic polynomial.
$endgroup$
– jnez71
Jan 23 at 18:25
$begingroup$
I dunno if fixed points in evolutionary models differ from the fixed points in ODE. We previously had a different model where the fixed point was calculated also as values $N^{*}$ which satisfy $N=f(N) cdot N$ since the original equation is $frac{dN}{dt}=f(N) cdot N$
$endgroup$
– Oliver
Jan 23 at 18:43