Citation: Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability[J]. Mathematical Biosciences and Engineering, 2017, 14(4): 933-952. doi: 10.3934/mbe.2017049
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Many natural phenomena can be formulated as the differential law of the development (evolution) in time of a physical system. The resulting differential equation combined with boundary conditions affecting the system are called evolution equations. Evolution equations are a popular framework for studying the dynamics of biological populations. For example, they have proven useful in understanding the dynamics of biological invasions [51], bacterial flocculation in activated sludge tanks [7], and the spread of parasites and diseases [30]. Since many biological populations converge to a time-independent state, many researchers have used theoretical tools to investigate long-term behavior of these models. Analytical and fixed point methods have been used to show the existence of stationary solutions to size-structured population models [44,29] and semigroup theoretic methods have been used to investigate the stability of these stationary solutions [28,43,4]. For many models in the literature, the principle of linearized stability [55,34] can be used to show that the spectral properties of the infinitesimal generator (IG) of the linearized semigroup determines the stability or instability of a stationary solution. Moreover, using compactness arguments, spectral properties of the infinitesimal generator can be determined from the point spectrum of the IG, which in turn can be written as the roots of a characteristic equation.
Despite this theoretical development, the derived existence and stability conditions are oftentimes rather complex, and accordingly the biological interpretation of these conditions can be challenging. To overcome this difficulty several numerical methods for stability analysis of structured population models have been developed [12,27,37,21]. For instance, Diekmann et al. [37,22,21] present a numerical method for equilibrium and stability analysis of physiologically structured population models (PSPM) or life history models, where individuals are characterized by a (finite) set of physiological characteristics (traits such as age, size, sex, energy reserves). In this method a PSPM is first written as a system of integral equations coupled with each other via interaction (or feedback) variables. Consequently, equilibria and stability boundary of the resulting integral equations are numerically approximated using curve tracing methods. Later, de Roos [19] presented the modification of the curve tracing approach to compute the demographic characteristics (such as population growth rate, reproductive value, etc) of a linear PSPM. For a fast and accurate software for theoretical analysis of PSPMs we refer interested reader to a software package by de Roos [20]. An alternative method for stability analysis of physiologically structured population models, developed by Breda and coworkers [11,13,10], uses a pseudospectral approach to compute eigenvalues of a discretized infinitesimal generator. This method (also known as infinitesimal generator (IG) approach) has been employed to produce bifurcation diagrams and stability regions of many different linear evolution equations arising in structured population modeling [13,14,15]. Unfortunately, not all structured population models fit into the framework of PSPMs and thus there is a need for a more general numerical framework for asymptotic analysis of structured population models.
In this paper we develop a numerical framework to guide theoretical analysis of structured population models. We demonstrate that our methodology can be used for numerical computation and stability analysis of positive stationary solutions of both linear and nonlinear size-structured population models. Moreover, we illustrate the utility of our framework to produce approximate existence and stability regions for steady states of size-structured population models. We also provide an open source Python program used for the numerical simulations in our GitHub repository [42]. Although, the examples presented in this paper are size-structured population models, in Section 2, we show that the framework is applicable to more general evolution equations.
The main idea behind the numerical framework is first to write a structured population model in the form of an evolution equation and then use the well-known Trotter-Kato Theorem [53,35] to approximate the infinitesimal generator of the evolution equation on a finite dimensional space. This in turn allows one to approximate solutions (or spectrum) of the evolution equation with the solution (or spectrum) of system of differential equations. Consequently, we approximate the stationary solutions of an actual model with stationary solutions of the approximate infinitesimal generator on a finite dimensional space. Approximate local stability of the approximate steady states are then computed from the spectrum of the Jacobian of ODE system evaluated at this steady states. Our method is similar to the IG approach in [13,14,15], in a sense that we also approximate infinitesimal generator and analyze the spectrum of the resulting operator to produce existence and stability regions. However, in contrast to IG approach, our framework also computes actual steady states and is easily applied to nonlinear evolution equations arising in structured population dynamics.
The rest of the paper is structured as follows. We describe the theoretical development of our framework for general evolution equations in Section 2.1. Note that readers with more biological background can skip Section 2.1 and directly jump into the application of the framework in Section 2.3. In Section 2.3, we illustrate the convergence of the approximation method by applying it to linear Sinko-Streifer model, for which the exact form of the stationary solutions is known. To further illustrate the utility of our approach, in Section 3, we apply our framework to a nonlinear size-structured population model (also known as population balance equations in the engineering literature) described in [5,9]. Moreover, in Section 2.2, we show that approximate local stability conditions for a stationary solution can be derived from the spectral properties of the approximate infinitesimal generator. Finally, we conclude with some remarks and a summary of our work in Section 5.
In this section, we demonstrate our numerical methodology for general evolution equations. We first present the numerical scheme used to approximate the infinitesimal generator of a semigroup. Subsequently, in Section 2.3, we illustrate the convergence of our approach by applying it to linear Sinko-Streifer equations, for which exact stationary solutions are known.
Given a Banach space
ut=F(u),u(0,∙)=u0∈X, | (1) |
where
u(t,x)=T(t)u0(x). |
The transition operator
1.
2.
3. For each fixed
limt→0+‖T(t)u0−u0‖=0. |
Moreover, showing that the operator
Let
‖πn‖≤˜M | (A1) |
for all
limn→∞πnv=v | (A2) |
for all
pnv=(α1,⋯,αn)T |
and the norm on
‖v‖Rn=‖p−1nv‖X. |
Consequently, we define bounded linear operators
Pnv=pnπnv,v∈X | (2) |
and
Enz=ιnp−1nz,z∈Rn, | (3) |
respectively. Finally, we define approximate operators
Fn(z)=PnF(Enz) | (4) |
for all
Accordingly, the Trotter-Kato Theorem states that the semigroup generated by the discrete operator
Theorem. (Trotter-Kato) Assume that (
‖T(t)‖X,‖Tn(t)‖Rn≤Mewtfor all t≥0,n∈N, |
for some constants
1. There exists a
‖En(λ0In−Fn)−1Pnv−(λ0I−F)−1v‖X→0as n→∞. |
2. For all
‖EnTn(t)Pnv−T(t)v‖X→0 |
as
In general, one establishes the first statement for a Trotter-Kato approximation and then uses the second statement to approximate an abstract evolution equation on a finite dimensional space. In their paper, Ito and Kappel [32] present the standard ways to establish the first statement of the theorem (see also [6,2,1]). Therefore, here we assume that for a particular problem the first statement in the theorem has already been established and thus the evolution equation in (1) can be approximated by the following system of ODEs,
u′n(t)=Fn(un(t)),un(0)=Pnu(0,∙). | (5) |
Consequently, the solution of the IVP is mapped onto the infinite dimensional Banach space
limn→∞‖Enun(t)−u‖X=0 | (6) |
for
In general, finding explicit stationary solutions of abstract evolution equations is a challenging task. Conversely, many efficient root finding methods have been developed for finding steady states of a system of ODEs. For large-scale nonlinear systems, many efficient methods have been developed as well. Hence, we propose a numerical framework that utilizes those efficient root finding methods to approximate steady state solutions of general evolution equations. The idea is to use an efficient and accurate root finding method to approximate a stationary solution of the evolution equation (1) with the stationary solutions of the IVP in (5). Thus, as a consequence of the Trotter-Kato Theorem, the steady states of (5) converge to the steady states of (1) as
Studying the asymptotic behavior of solutions is a fundamental tool for exploring the evolution equations which arise in the mathematical modeling of real world phenomena. To this end, many mathematical methods have been developed to describe long-term behavior of evolution equations. For instance, for long-time behavior of linear evolution equations, linear semigroup theoretic methods can be used to formulate physically interpretable conditions. Furthermore, for nonlinear evolution equations, the principle of linearized stability can be used to relate the spectrum of the linearized infinitesimal generator to the local stability or instability of the stationary solution. Nevertheless, investigating the spectrum of the linearized infinitesimal generator is cumbersome and requires advanced functional analysis techniques. In contrast to general evolution equations, the asymptotic behavior of ordinary differential equations are determined by the eigenvalues of the Jacobian and well-studied. Hence, in this section we demonstrate that the approximation scheme, presented in Section 2.1, can also be used to give some insights about the stability of stationary solutions of the general evolution equations.
Stability results discussed in this section are not in a traditional Lyapunov sense. In particular, since stationary solutions discussed in this paper are only approximations to actual stationary solutions, the stability results only hold for finite time intervals. Therefore, we refer to this kind of stability as approximate local stability of stationary solutions as this stability is deduced from numerical approximation of an evolution equation. In mathematical terms, the local numerical stability of a stationary solution is defined as follows.
Definition 2.1. Stationary solution
‖u(t,⋅)−u∗‖X<ϵ | (7) |
for all
Having the required definition in hand, we now prove the following stability result for general evolution equations.
Corollary 2.2. Let
Proof. Since the infinitesimal generator approximation scheme, presented in Section 2.1, is convergent, for every given
‖u(t,⋅)−Enun(t)‖X<ϵ/2 | (8) |
for all
‖uM(t,⋅)−PMu∗‖RM=‖EMuM(t,⋅)−u∗‖X<ϵ/2 | (9) |
for all
‖u(t,⋅)−u∗‖X≤‖u(t,⋅)−EMuM(t,⋅)‖X+‖EMuM(t,⋅)−u∗‖X<ϵ |
for all
We note that although the stability result of Corollary 2.2 holds for arbitrarily large finite time intervals, the Corollary does not guarantee Lyapunov stability of stationary solutions.
To verify convergence of the proposed approximation scheme, we apply the framework to the linear Sinko-Streifer model [52] for which an exact form of the stationary solution is available. The model describes the dynamics of single species populations and takes into account the physiological characteristics of animals of different sizes (and/or ages). The mathematical model reads as
ut=G(u)=−(gu)x−μu,t≥0,0≤x≤¯x<∞ | (10) |
with a McKendrick-von Foerster type renewal boundary condition at
g(0)u(t,0)=∫¯x0q(y)u(t,y)dy |
and initial condition
u(0,x)=u0(x). |
The variable
Setting the right side of the equation (10) to zero and integrating over the size on
u∗(x)=1g(x)exp(−∫x0μ(s)g(s)ds)∫¯x0q(y)u∗(y)dy. | (11) |
Multiplying both sides of (11) by
1=∫¯x0q(x)g(x)exp(−∫x0μ(s)g(s)ds)dx. | (12) |
The convergence of the approximation scheme presented in Section 2.1 for Sinko-Streifer models has already been established in [6]. Using the basis functions for
βni(x)={1;xni−1<x≤xni;i=1,…,n0;otherwise |
for positive integer
Xn={h∈X|h=n∑i=1αiβni,αi∈R}, |
and accordingly, we define the orthogonal projections
πnh(x)=n∑j=1αjβnj(x),whereαj=1Δx∫xnjxnj−1h(x)dx. |
Moreover, since the evolution equation defined in (10) is a linear partial differential equation, the approximate operator
[Gn]ij={−1Δxg(xni)−μ(xni)+q(xnj)i=j=1q(xnj)i=1,j≥2−1Δxg(xni)−μ(xni)i=j≥21Δxg(xni−1)i=j+1,j≥10otherwise. | (13) |
At this point, one can use numerical techniques to calculate zeros of the linear system
Gnun=0. | (14) |
For the purpose of illustration, we set the model rates to
q(x)=a(x+1),g(x)=b(x+1),μ(x)=c. | (15) |
Plugging this rates into the necessary condition (12) yields
a={ln2bb=c(b−c)/(21−c/b−1)b≠c. |
For
For the purpose of illustration, we arbitrarily choose
In aerosol physics and environmental sciences, studying the flocculation of particles is widespread. The process of flocculation involves disperse particles in suspension combining into aggregates (i.e., a floc) and separating. The mathematical model used to study flocculation process is the well-known population balance equation (PBE) which describes the time-evolution of the particle size number density. The equations for the flocculation model track the time-evolution of the particle size number density
ut=F(u) | (16) |
where
F(u):=G(u)+A(u)+B(u), |
G(u):=−∂x(gu)−μ(x)u(t,x), | (17) |
A(u):=12∫x0ka(x−y,y)u(t,x−y)u(t,y)dy−u(t,x)∫ˉx−x0ka(x,y)u(t,y)dy, | (18) |
and
B(u):=∫¯xxΓ(x;y)kf(y)u(t,y)dy−12kf(x)u(t,x). | (19) |
The boundary condition is traditionally defined at the smallest size
g(0)u(t,0)=∫¯x0q(x)u(t,x)dx,u(0,x)=u0(x)∈L1(0,¯x), |
where the renewal rate
∫y0Γ(x;y)dx=1 for all y∈(0,¯x]. | (20) |
The population balance equation, presented in (16), is a generalization of many mathematical models appearing in the size-structured population modeling literature and has been widely used, e.g., to model the formation of clouds and smog in meteorology [49], the kinetics of polymerization in biochemistry [56], the clustering of planets, stars and galaxies in astrophysics [39], and even schooling of fish in marine sciences [47]. For example, when the fragmentation kernel is omitted,
The equation (16) has also been the focus of considerable mathematical analysis. Well-posedness of the general flocculation model was first established by Ackleh and Fitzpatrick [1,2] in an
For the numerical implementation we adopt the scheme developed in Section 2.1. Therefore, the approximate formulation of (16) becomes the following system of
˙un=Fn(un)=Gnun+PnA(Enun)+PnB(Enun), | (21) |
un(0,x)=Pnu0(x), | (22) |
where the matrix
PnA(Enun)=(−α1n−1∑j=1ka(xn1,xnj)αjΔx12ka(xn1,xn1)α1α1Δx−α2n−2∑j=1ka(xn2,xnj)αjΔx⋮12n−2∑j=1ka(xnj,xnn−1−j)αjαn−1−jΔx−αn−1ka(xnn−1,xn1)α1Δx12n−1∑j=1ka(xnj,xnn−j)αjαn−jΔx) |
and
PnB(Enun)=(n∑j=2Γ(xn1;xnj)kf(xnj)αjΔx−12kf(xn1)α1n∑j=3Γ(xn2;xnj)kf(xnj)αjΔx−12kf(xn2)α2⋮Γ(xnn−1;xnn)kf(xnn)αnΔx−12kf(xnn−1)αn−1−12kf(xnn)αn). |
The convergence of the approximate scheme (21)-(22) has been established in [1]. Therefore, the stationary solutions of the microbial flocculation model (16) can be systematically approximated by the stationary solutions of the system of nonlinear ODEs given in (21). We used Powell's hybrid root finding method [48] as implemented in Python 2.7.101 to find zeros of the steady state equation (see available code at [42]). For faster convergence rate, we provided the solver with the exact Jacobian of
Γ(x,y)=1[0,y](x)6x(y−x)y3, |
1 scipy.optimize.fsolve
2 Although normal and log-normal distributions are mostly used in the literature, Byrne et al. [16] have provided evidence that the Beta density function describes the fragmentation of small bacterial flocs.
where
ka(x,y)=(x1/3+y1/3)3 |
Other model rates were chosen arbitrarily as
q(x)=a(x+1),g(x)=b(x+1)μ(x)=cxkf(x)=x, |
where
The main advantage of this approximation scheme (21)-(22) is that it can be initialized very fast using Toeplitz matrices [40]. Fast initialization of the discretization scheme allows one to check the existence of the steady states at many discrete points efficiently. This in turn allows for the generation of the existence and stability regions of the steady states of the PBE in (16). To illustrate the existence regions of the steady states of the PBE, we discretized the intervals
{u0(x)=2i|i=0,1,…,9}, |
before we conclude a positive steady state does not exist for a given point
Figure 3a illustrates an example stationary solution for
M0(t)=∫ˉx0u(t,x)dx≈n∑i=1∫xnixni−1αiβni(x)dx=Δxn∑i=1αi, |
and total mass of the flocs (first moment),
M1(t)=∫ˉx0xu(t,x)dx≈n∑i=1∫xnixni−1αixβni(x)dx=Δx2n∑i=1αi(xni+xni−1). |
Moreover, to confirm that the steady state solution is not changing with increasing dimension of approximate spaces
In this section, we derive conditions for approximate local stability of the stationary solution of the nonlinear population balance equation defined in (16). In particular, we impose conditions on the model rates of the population balance equation for which the first statement of Corollary 2.2 holds. Towards this end, we use the well-known Gershgorin theorem for locating eigenvalues of a matrix. The Gershgorin theorem states that each eigenvalue of
{z∈C:|z−aii|≤Ri}, |
where
Ri=n∑j=1,j≠i|aji|. |
Since the approximate system for the microbial flocculation model is nonlinear, we linearize the system around its stationary solutions. Let
JF(α)=Gn+JA(α)+JB(α), | (23) |
where
JA(α)=(−α1ka(xn1,xn1)Δx−α1ka(xn1,xn2)Δx⋯−α1ka(xn1,xnn−1)Δx0−α2ka(xn2,xn1)Δx⋯−α2ka(xn2,xnn−2)Δx00⋮⋮⋱⋮⋮−αN−1ka(xnn−1,xn1)Δx0⋯0000⋯00)+(−n−1∑j=1ka(xn1,xnj)αjΔx0⋯00α1ka(xn1,xn1)Δx−n−2∑j=1ka(xn2,xnj)αjΔx0⋯0α2ka(xn1,xn2)Δxα1ka(xn2,xn1)Δx⋱0⋮⋮⋮⋱−1∑j=1ka(xnn−1,xnj)αjΔx0αn−1ka(xn1,xnn−1)Δxαn−2ka(xn2,xnn−2)Δx⋯α1ka(xnn−1,xn1)Δx0), |
and
JB(α)=(−12kf(xn1)Γ(xn1;xn2)kf(xn2)ΔxΓ(xn1;xn3)kf(xn3)Δx⋯Γ(xn1;xnn)kf(xnn)Δx0−12kf(xn2)Γ(xn2;xn3)kf(xn3)Δx⋯Γ(xn2;xnn)kf(xnn)Δx⋮0⋱⋱⋮0⋯0−12kf(xnn−1)Γ(xnn−1;xnn)kf(xnn)Δx00⋯0−12kf(xnn)) |
To bound the eigenvalues of
aii=−1Δxg(xni)−μ(xni)−12kf(xni)−αika(xni,xni)Δx−n−i∑j=1ka(xni,xnj)αjΔx |
and
Ri≤1Δxg(xni)+q(xni)+i−1∑j=1Γ(xnj;xni)kf(xni)Δx+n−i∑j=1αjka(xnj,xni)Δx |
+n−i∑j=1,j≠iαjka(xni,xnj)Δx, |
respectively. Consequently, if we can show that
|aii|>Rifor each i∈{1,…,n}, | (24) |
then each of the Gershgorin disks lie strictly on the left side of the complex plane. To this end, inequality (24) can be simplified as
μ(xni)+12kf(xni)>q(xni)+i−1∑j=1Γ(xnj;xni)kf(xni)Δx+n−i∑j=1αjka(xnj,xni)Δx | (25) |
for each
μ(x)+12kf(x)>q(x)+∫x0Γ(y,x)kf(x)dy+∫¯x−x0ka(x,y)u∗(y)dy | (26) |
for all
q(x)+12kf(x)−μ(x)+∫¯x−x0ka(x,y)u∗(y)dy<0 |
for all
Proposition 3.1. Let
q(x)+12kf(x)−μ(x)+∫¯x−x0ka(x,y)u∗(y)dy<0 | (27) |
for all
∂y(ka(x,y)u∗(y))≤0 and ∂yΓ(y,x)≤0 | (28) |
for all
To illustrate the utility of the framework developed in this section we applied our approach to the model rates given in Section 3.1 for generation of Figure 3a. The Beta distribution used for the post-fragmentation function
Our primary motivation in this paper was to show that available numerical tools in the literature can facilitate theoretical analysis of evolution equations. Towards this end we developed a numerical framework for theoretical analysis of evolution equations arising in population dynamical models. The main idea behind this framework is to approximate generators of semigroups of evolution equations and use numerical tools to study stability of steady states of evolution equations. We demonstrated the utility of our approach by applying the numerical framework to both linear and nonlinear size-structured population models. In particular, we generated approximate existence and stability regions of the steady states of both models (which can be difficult to obtain by using conventional analytical tools). We showed that our numerical framework can also be used to gain insight about the approximate local stability (see Definition 2.1) of stationary solutions. Furthermore, code used for the numerical simulations can be obtained from our GitHub repository under the open source MIT License (MIT) [42].
Although the stability inequality in (7) holds for arbitrarily large finite time intervals, behavior of the solutions as
Funding for this research was supported in part by grants NSF-DMS 1225878 and NIH-NIGMS 2R01GM069438-06A2. This work utilized the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-0821794) and the University of Colorado Boulder. The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver and the National Center for Atmospheric Research.
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