Linearization around the equilibrium point for a system of differential equationsdNdt=μN(1−NK)+N∫0∞p(a,t)da∂p∂t+∂p∂a=kp(a,t).Linearizing it around (N=K,p=0), I am getting the systemdNdt=h−μN+N∫0∞p(a,t)da∂p∂t+∂p∂a=kp(a,t),where h=μK.The reason I am getting this system is because I am considering N∫0∞p(a,t)da as linear term. Is this linearisation right or the correct system is dNdt=h−μN∂p∂t+∂p∂a=kp(a,t),where h=μK.

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Laylah Hebert · Accepted Answer

Step 1I agree with the linearisation of the second equation. For the first, you are close but not quite correct. To find the linearisation around&amp;nbsp;N=K,p=0, substitute&amp;nbsp;N=K+εN~&amp;nbsp;and&amp;nbsp;p=εp~&amp;nbsp;into your equations and throw out everything that isn&#039;t order&amp;nbsp;ε. I got:εdN~dt=dNdt=μN(1−NK)+N∫0∞p(a,t)&amp;nbsp;da&amp;nbsp;=−μ(K+εN~)ε(N~K)+ε(K+εN~)∫0∞p~(a,t)&amp;nbsp;da&amp;nbsp;=(−μN~+K∫0∞p~(a,t)&amp;nbsp;da)ε+o(ε).Step 2Thus, the linearised equation is (dropping the tildes):dN&amp;nbsp;dt&amp;nbsp;=−μN+K∫0∞p(a,t),da.Edit: Answer to question in the comments. Let&amp;nbsp;N⋅,p⋅&amp;nbsp;be steady state solutions to your system. Replacing p with&amp;nbsp;p∗+εp&amp;nbsp;the equation for p becomes:kp∗+εkp=k(p∗+εp)=∂p∗∂t+ε∂p∂t+∂p∗∂a+ε∂p∂a&amp;nbsp;=∂p∗∂a+ε(∂p∂t+∂p∂a)which implies the linearised equation iskp=∂p∂t+∂p∂a(not surprising since the original equation was linear). For the equation for N replace N and p with&amp;nbsp;N∗+εN&amp;nbsp;and&amp;nbsp;p∗+εp&amp;nbsp;respectively:Step 3εdNdt=d(N∗+εN)dt&amp;nbsp;=μ(N∗+εN)(1−N∗+εNK)+(N∗+εN)∫0∞(p∗(a)+εp(a,t))&amp;nbsp;da&amp;nbsp;=μN∗(1−N∗K)+N∗∫0∞p∗&amp;nbsp;da&amp;nbsp;&amp;nbsp;+ε[N∗∫0∞p&amp;nbsp;da+N∫0∞p∗&amp;nbsp;da+μN(1−2N∗K)]&amp;nbsp;&amp;nbsp;+ε2[N∫0∞p&amp;nbsp;da−μN2K]&amp;nbsp;=ε[N∗∫0∞p&amp;nbsp;da+N∫0∞p∗&amp;nbsp;da+μN(1−2N∗K)]&amp;nbsp;&amp;nbsp;+ε2[N∫0∞p&amp;nbsp;da−μN2K]using that&amp;nbsp;(N∗,p∗)&amp;nbsp;is the steady state solution. Thus, the linearised equation is:dN&amp;nbsp;dt&amp;nbsp;=N∗∫0∞p,da+N∫0∞p∗,da+μN(1−2N∗K).

Linearization around the equilibrium point for a system

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