I have this population model here:
\begin{align} \begin{cases} \frac{\partial}{\partial t} p(t,a,x) + \frac{\partial}{\partial a} (\Gamma_0 p(t,a,x)) + \frac{\partial}{\partial x} (\Gamma_1(a,x) p(t,a,x)) \\\ = -(L(a,x) + F(a,x) + d_1) p(t,a,x) + G(N(t))q(t,a,x),\\ \frac{\partial}{\partial t} q(t,a,x) = L(a,x) p(t,a,x) - (G(N(t)) + d_2)q(t,a,x). \end{cases} \label{Hauptmodell Grundformel} \end{align} with initial conditons \begin{align} p(0,a,x) = p_i(a,x), \;\; a \geq 0, \,x \geq 0, \label{Hauptmodell Anfangsbedingung p} \end{align} and \begin{align} q(0,a,x) = q_i(a,x), \;\; a \geq 0, \,x \geq 0, \label{Hauptmodell Anfangsbedingung q} \end{align} and \begin{align} p(t,0,x) = \frac{2}{\Gamma_0} \int_0^\infty \int_0^\infty f(a,x,y) \, p(t,a,y) \, da \, dy. \end{align}
and get this eigenvalue problem
\begin{align} \begin{cases} \lambda P + \frac{\partial (\Gamma_0 P)}{\partial a} + \frac{\partial (\Gamma_1(a,x) P) }{\partial x} = -(L(a,x) + F(a,x) + d_1) P + \tilde{G}Q,\\ (\lambda + \tilde{G} + d_2)Q = L(a,x) P,\\ P(0,x) = \frac{2}{\Gamma_0} \int_0^\infty \int_0^\infty f(a,x,y) \, P(a,y) \, da \, dy. \end{cases} \label{stationäre Lösung} \end{align}
and then this dual/ adjoint problem \begin{align} \begin{cases} \lambda \varphi - \Gamma_0 \frac{\partial \varphi}{\partial a} - \Gamma_1(a,x) \frac{\partial \varphi}{\partial x} -2\int_0^\infty \varphi(0,y) f(a,y,x) \,dy\\ = -(L(a,x) + F(a,x) + d_1) \varphi + L(a,x) \psi,\\ (\lambda + \tilde{G} + d_2)\psi = \tilde{G} \varphi \label{adjungiertes System} \end{cases} \end{align}
with normalisation \begin{align*} \int_0^\infty \int_0^\infty \left(\varphi(a,x) P(a,x) + \psi(a,x) Q(a,x)\right)\,da\,dx = 1. \end{align*}.
And now they say that solutions of the normal population model in the beginning satisfies this condition here: \begin{align} \int_0^\infty \int_0^\infty \left(\varphi(a,x)p(t,a,x) + \psi(a,x)q(t,a,x)\right)\,da\,dx\\ = e^{\lambda t}\int_0^\infty \int_0^\infty \left(\varphi(a,x)p_i(a,x) + \psi(a,x)q_i(a,x)\right)\,da\,dx \end{align}
Can someone explain how to get this condition in the end?