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A numerical framework for computing steady states of structured population models and their stability

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  • Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can \emph{also} be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (github.com/MathBioCU).

    Mathematics Subject Classification: Primary: 35B40, 92B05; Secondary: 65N40.


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  • Figure 1.  Results-of-the Results of the numerical simulations. a) $a,\,b\,,c$ values satisfying the necessary condition (12), form a 3D surface (blue surface). Steady states of the Sinko-Streifer model only exist on the red line b) Comparison of exact stationary solution (for the point marked with red star in Figure 1a) with approximate stationary solution for $n=100$. c) Absolute error between exact stationary solution and approximate stationary solution decays linearly as the dimension of approximate subspaces $\mathcal{X}_{n}$ increase.

    Figure 2.  Existence and stability regions for the steady states of the PBE a) Existence region for the steady states of the PBE forms a wedge like shape. b) Stability region for $b=0.1$, $a\in[0,\,15]$ and $c\in[0,\,5]$. c) Stability region for $b=0.5$, $a\in[0,\,15]$ and $c\in[0,\,5]$. d) Stability region for $b=1.0$, $a\in[0,\,15]$ and $c\in[0,\,5]$. Color bar represents the real part of rightmost eigenvalue of the Jacobian matrix evaluated at each steady state. Yellow regions represents the region for which a positive steady state does not exists.

    Figure 3.  An example steady-state solution of the PBE for $b=0.5,\,a=c=1$. b) Steady states for increasing renewal rate and $b=c=1$

    Figure 4.  Time evolution of the flocculation model with arbitrary initial conditions. a) Four different initial conditions are chosen close to the steady state. b) Solution of the PBE for those initial conditions at $t=10$. c) Evolution of the total number $M_{0}(t)$ of the flocs for $t\in[0,\,10]$. d) Evolution of the total mass $M_{1}(t)$ of the flocs for $t\in[0,\,10]$.

    Figure 5.  Change in zeroth and first moments with increasing dimension of the approximate space $\mathcal{X}_{n}$. a) Change in the total number and the total mass of the flocs with respect to increasing dimension $n$. Dashed red lines and dotted green lines corresponds to the total number and the total mass of the flocs of the steady state for $n=1000$, respectively. b) Steady state solution for $n=100$ and $n=500$.

    Figure 6.  Eigenvalues of the Jacobian $J_{\mathcal{F}}(\alpha)$ multiplied by $\Delta x$ for the steady state illustrated in Figure 3a. a) Eigenvalues of the Jacobian plotted in the complex plane for $n=20$. b) Eigenvalues of the Jacobian plotted in the complex plane for $n=50$. c) Eigenvalues of the Jacobian plotted in the complex plane for $n=200$. d) Change in the rightmost eigenvalue for increasing $n$. Dashed black line corresponds to the rightmost eigenvalue of the Jacobian for $n=1000$.

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