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Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods
Abstract
This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.
... VOLUME 6, 2018 Note that a summary of pseudospectral discretization methods according to (9)- (14), (29) can be found e.g. in the book by Breda et al. [91], and these methods were sufficiently applied to biological models (see e.g. [92]). The extension to inter alia systems with uncertain parameters was presented in [93]. ...
... Let us name just a few applications, besides the already introduced ones [104], [218], [223]. Breda et al. [92] dealt with physiologically structured populations of the Daphnia type. The authors revisited the pseudospectral approach [93] to compute the eigenvalues of the infinitesimal generator of linearized systems modeled by the Volterra functional equation and the FDEs, to study the local asymptotic stability of equilibria and relevant bifurcations. ...
... Lehotzky and Insperger [48] computed the rightmost part of the spectrum for distributeddelay systems. Breda et al. [92] and Beretta and Breda [237] utilized the pseudospectral approach to determine stability of population models. Kammer and Olgac [169] solved the DDS problem for dRTDSs via the CTCR, or it is worth highlighting the work of Michiels and Ünal [130] on FIR filters, to name just a few. ...
In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampleddata, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles – which are most closely related to the research area – are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic; Second, to suggest possible future research directions to be tackled by scientists and engineers in the field.
... Yet it is fair to say that at present no toolbox exists for showing numerically that a nonlinear renewal equation exhibits a rich repertoire of dynamical behavior. Recently, a new branch of research has emerged, addressing the numerical stability and bifurcation properties of delay equations (renewal equations and delay differential equations) through their pseudospectral approximation, see, e.g., [3][4][5][6][7][8][9]. Initially the focus was on linear problems, but now it is extended to nonlinear problems, aiming at providing tools for the systematic discretization and analysis of general delay equations. ...
... Nevertheless, some effort by the authors and colleagues in this direction is currently ongoing, targeted to specific objectives in the ample framework of continuation techniques. See, e.g., [4] for the stability analysis of equilibria and [47] for more general aspects of pseudospectral methods in the context of structured population models. ...
... Concerning the renewal equations treated here, steady states can be found analytically, so that one only needs to approximate the associated characteristic roots. These can be obtained as eigenvalues of the corresponding infinitesimal generator, via the pseudospectral techniques in [3,4]. Both reduce the generator to a finite-dimensional matrix, whose eigenvalues are taken as approximations. ...
We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits transcritical, Hopf and period doubling bifurcations. The reliability is demonstrated by comparing the results to those obtained by a reduction to a Hamiltonian Kaplan–Yorke system and to those obtained by direct application of collocation methods (the latter also yield estimates for positive Lyapunov exponents in the chaotic regime). We conclude that the methodology described here works well for a class of delay equations for which currently no tailor-made tools exist (and for which it is doubtful that these will ever be constructed).
... Different numerical methods have been proposed for the stability analysis of equilibria of DDEs (see, e.g., [9,32] and the references therein) and coupled REs/DDEs [13]. Here, in particular, we focus on the so-called IG-approach developed for linear DDEs, REs, and coupled REs/DDEs [4,5,6,7,8,9]. It is based on the pseudospectral method [29,41] to discretize the infinitesimal generator associated with linear(ized) delay equations. ...
... The basic results for such equations, as well as for coupled REs/DDEs, have been developed in [17], making available all the key ingredients to extend the pseudospectral discretization also to the nonlinear RE (3.1), i.e., the well-posedness of the associated initial value problem, the abstract formulation, the principle of linearized stability, and the bifurcation theorems. The last ingredient to complete the program, i.e., the theorem on the convergence of the eigenvalues of the discretized generator to the eigenvalues of the original one in the linear case, can be obtained from [5]. We remark that the same convergence problem is addressed in [7] for a different choice of the underlying state space, namely, X = C([−τ, 0]; R d ). ...
... In particular, x is locally asymptotically stable if the zero solution of (3.14) is exponentially stable, and it is unstable if the zero solution of (3.14) is unstable. The theorem concerning the spectral accuracy of the IG-approach in approximating the eigenvalues of the infinitesimal generator of linear REs, which is analogous to Theorem 2.6, has been proved in [5]. Then, by the same arguments as used in section 2.2, we conclude the validity of the appropriate analogues of Theorem 2.5, concerning the commutativity between pseudospectral discretization and linearization, and of Corollary 2.7, ensuring that the study of the approximating ODE (3.10) gives accurate information about the stability and bifurcation of the equilibria of the nonlinear RE (3.1). ...
We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations.
... For the analysis of local stability of equilibria, pseudospectral methods have been widely used both for delay equations [10][11][12][13] and for PDE population models with one structuring variable [14][15][16]. The main advantage of pseudospectral methods is their typical spectral accuracy, by which the order of convergence of the approximation error increases with the regularity of the approximated function. ...
... To further illustrate the efficacy of the method on realistic models with nontrivial velocity, we consider a model inspired from ecology, with individuals structured by their demographic age a and their size z, which grows in time with velocity g(z) [7]. We will take parameters inspired by the Daphnia population growth model [12,39], but assuming that the algae resource is fixed at a certain value. We consider a slight modification of parameters so that both the survival probability at the maximal age and the growth rate at the maximal size are zero. ...
The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.
... For a reference literature see [11,12,13,17,18]. The following description is mainly inspired by [7]. ...
... Most of the model parameters are hidden in (13), appearing only in the definitions of the various rates defining the model. In this regard we refer to [7] for the current choices and, for the reader's convenience, we recall from there all the necessary quantities in Table 1. ...
... For this reason, in the last years the interest in the study of the dynamics of delay models has been increasing and important challenges, in particular numerical, have been identified. Indeed, delay equations describe infinite-dimensional dynamical systems, and theoretical results should be complemented with efficient numerical methods to approximate solutions of initial value problems [3,4,5,7,17,15,16,19,44,43], boundary value problems [48,49,50], and to investigate the stability of equilibria and periodic solutions [10,11,12,13,14,47,52,53,66]. In applications the attention is focused not only on the approximation of the dynamical properties for some given parameter values, but also on how such properties change when varying some parameters. ...
... Indeed, thanks to the integral equations, we can relate the convergence of the nonlinear semigroups to the convergence of the trivial linear semigroups T 0,M (t) to T * 0 (t). Moreover (4) and (20) suggest that the matrix (16) "mimics" the infinitesimal generator A * 0 of the semigroup {T * 0 (t)} t≥0 associated to the trivial DDE (10). Therefore, to understand the dynamical behavior of (18), it is crucial to investigate the properties of the matrix A 0,M and of the corresponding (20) it is clear that the matrix D M plays a major role. ...
... For a reference literature see [11,12,13,17,18]. The following description is mainly inspired by [7]. ...
... Most of the model parameters are hidden in (13), appearing only in the definitions of the various rates defining the model. In this regard we refer to [7] for the current choices and, for the reader's convenience, we recall from there all the necessary quantities in Table 1. ...
Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.
... The perhaps most generally applicable method for stability analysis in continuous population dynamics is the pseudospectral discretisation approach. The approach was first applied for studying the stability of the zero solution of linear DDE with fixed delay by Breda et al. (2005Breda et al. ( , 2013, see also Breda et al. (2015b) for a review, and applied to the linearisation of a size-structured consumer-resource model by Breda et al. (2015a). The pseudospectral discretisation was then applied by Breda et al. (2016a) to nonlinear delay equations, including both differential and integrodifferential equations with fixed delay, and later extended to equations with infinite delay by Gyllenberg et al. (2018). ...
... Since the eigenfunctions associated with the linearised system are exponential, we can exploit the spectral convergence of the interpolation process and use the argument of Breda et al. (2005) to ensure the convergence of the associated characteristic roots. In particular, the eigenvalues corresponding to the linearisation of (7.6) approximate the rightmost eigenvalues of the linearisation of (7.1) with spectral accuracy, as proven by Breda et al. (2005Breda et al. ( , 2015a. In conclusion, from Theorem 2.4 and Corollary 2.7 of Breda et al. (2016a), by using results on interpolation on [−h, 0], the following result follows. ...
We consider a mathematical model describing the maturation process of stem cells up to fully mature cells. The model is formulated as a differential equation with state-dependent delay, where maturity is described as a continuous variable. The maturation rate of cells may be regulated by the amount of mature cells and, moreover, it may depend on cell maturity: we investigate how the stability of equilibria is affected by the choice of the maturation rate. We show that the principle of linearised stability holds for this model, and develop some analytical methods for the investigation of characteristic equations for fixed delays. For a general maturation rate we resort to numerical methods and we extend the pseudospectral discretisation technique to approximate the state-dependent delay equation with a system of ordinary differential equations. This is the first application of the technique to nonlinear state-dependent delay equations, and currently the only method available for studying the stability of equilibria by means of established software packages for bifurcation analysis. The numerical method is validated on some cases when the maturation rate is independent of maturity and the model can be reformulated as a fixed-delay equation via a suitable time transformation. We exploit the analytical and numerical methods to investigate the stability boundary in parameter planes. Our study shows some drastic qualitative changes in the stability boundary under assumptions on the model parameters, which may have important biological implications.
... While working on their paper (Breda et al. 2015), Philipp Getto and Julia Sánchez Sanz found that formula (4.22) is incomplete (indeed, the solution operator of the homogeneous version of the linearized equation (4.3) should act on the second term at the right hand side of (4.22)). As a consequence (4.23), (4.24), (4.25), (4.31) and (4.32) are incorrect as well. ...
... We also thank Francesca Scarabel for checking the correctness of the formulas above. Moreover, we are very happy with the continuation of our work in Breda et al. (2015). ...
... That is, we focus here on a class of structured consumer-resource models, which describe the interaction and population dynamics of a size-structured consumer and its unstructured resource. On the one hand there is significant intrinsic mathematical interest in these nonlinear models, as they pose analytical and computational challenges (see for example [3,16,20,22,23,24,32,33]). On the other hand, for particular choices of the model ingredients, they are also used to investigate or demonstrate the richness of the dynamical behaviour of, for instance, a size-structured population of Daphnia feeding on algae (see e.g. ...
To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first-order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper, we delineate in what sense the two semigroups are equivalent. In particular, we (i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, (ii) focus on large time behavior and (iii) consider full orbits, i.e. orbits defined for time running from [Formula: see text] to [Formula: see text]. Conceptually, the PDE formulation is by far the most natural one. It has, however, the technical drawback that the solution operators are not differentiable, precluding rigorous linearization. (The underlying reason for the lack of differentiability is exactly the same as in the case of state-dependent delay equations: we need to differentiate with respect to a quantity that appears as argument of a function that may not be differentiable.) For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearized Stability. Next, the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation.
... For the analysis of local stability of equilibria, pseudospectral methods have been widely used both for delay equations [6,7,9,14] and for PDE population models with one structuring variable [4,10,34]. The main advantage of pseudospectral methods is their typical spectral accuracy, by which the order of convergence of the approximation error increases with the regularity of the approximated function. ...
The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. We propose an equivalent reformulation of the problem in the space of absolutely continuous functions in the sense of Carath\'eodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.
... That is, we focus here on a class of structured consumer-resource models, which describe the interaction and population dynamics of a size-structured consumer and its unstructured resource. On the one hand there is significant intrinsic mathematical interest in these nonlinear models, as they pose analytical and computational challenges (see for example [3,16,20,22,23,24,32,33]). On the other hand, for particular choices of the model ingredients, they are also used to investigate or demonstrate the richness of the dynamical behaviour of, for instance, a size-structured population of Daphnia feeding on algae (see e.g. ...
To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper we delineate in what sense the two semigroups are equivalent. In particular, we i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, ii) focus on large time behaviour and iii) consider full orbits, i.e., orbits defined for time running from to . For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearised Stability. Next the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation.
... We stress however that the extension to systems of equations is quite straightforward and can be done along the lines of [4]. In fact, the combination of the current method for RE with the pseudospectral discretization of DDE [4,38] provides a strategy for approximating general systems where a RE is coupled with a DDE, which arise frequently in population dynamics, see e.g. [3]. ...
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system can be about ten times more efficient in terms of computational times than the one originally proposed in Breda et al. (2016), as it avoids the numerical inversion of an algebraic equation.
... We stress however that the extension to systems of equations is quite straightforward and can be done along the lines of [4]. In fact, the combination of the current method for RE with the pseudospectral discretization of DDE [4,38] provides a strategy for approximating general systems where a RE is coupled with a DDE, which arise frequently in population dynamics, see e.g. [3]. ...
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the pseudospectral approach to the abstract formulation of the differential equation, we obtain an approximating system of ordinary differential equations. We present convergence proofs for equilibria and the associated characteristic roots, and we use some models from ecology and epidemiology to illustrate the benefits of the approach to perform numerical bifurcation analyses of equilibria and periodic solutions. The numerical simulations show that the implementation of the new approximating system is ten times more efficient than the one originally proposed in [Breda et al, SIAM Journal on Applied Dynamical Systems, 2016], as it avoids the numerical inversion of an algebraic equation.
... Nevertheless, the present work offers a first, solid background to start elaborating a succeeding strategy towards the proof of convergence. The authors plan to make this effort in the immediate future, also to substantiate the encouraging experimental results already obtained by extending the piecewise collocation to renewal equations, even coupled to RFDEs (for the target class of realistic models we have in mind see, e.g., [12,14,36]). ...
We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math. (2016) 133(3):525-555], [S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2771--2793] and [S. Maset, SIAM J. Numer. Anal. (2015) 53(6):2794--2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable of such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical being it directly linked to the course of time. Finally, we prove that the finite element method is convergent, while limit ourselves to comment on the unfeasibility of this approach as far as the spectral element method is concerned.
... Of course, numerical approximation is needed, as they are operators on Banach spaces of functions. In this regard, we rely on the pseudospectral methods in either [35] or [32] depending on the chosen formulation, both furnishing accurate approximations of the rightmost eigenvalue, which is the stability determining one. The results are illustrated in Figure 4, which is the analogous of Figure 2 but for Eq (2.6) replacing Eq (2.5). ...
A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents' decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.
... For this reason the system is approximated using the piecewise approach described in Section 2.2. Figure 1 shows the output of the continuation of the existence and stability boundaries of the nontrivial equilibrium in the plane (μ, K), for different values of the discretization index M. The stability boundaries agree with those obtained in [6,10] using specific numerical continuation algorithms. ...
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.
... we define the matrix Σ(E) ∈ R M ×M by 110 Σ kj (E) := x k xb µ(y, E) j (y) dy, j, k = 1, . . . , M,(2.6) and the diagonal matrix G(E) ∈ R M ×M with diagonal entries equal to G kk (E) = g(x k , E). ...
Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; the result is also proved theoretically under some regularity conditions on the model ingredients.
... In the last couple of decades, for instance, the theory of strongly continuous semigroups has furnished a solid theoretical background to most numerical methods devoted to the approximation of a (dominant or rightmost) part of the spectrum, see, e.g., [10,12,15,23] to name just a few specific techniques, or [13,40] for a compendium. Nevertheless, the approach through the characteristic equation is still extensively adopted, although necessarily restricted to selected, mainly low-dimensional, yet important and interesting models. ...
Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubtedly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equilibria inherit the stability properties of the corresponding linearizations, the study of which relies on associated characteristic equations. The transcendence of the latter, due to the presence of the delay, leads to infinitely-many roots in the complex plane. Simple algebraic manipulations show, first, that all such roots belong to the intersection of two curves. Second, only one of these curves is crucial for stability, and relevant sufficient and/or necessary criteria can be easily derived from its analysis. Other aspects can be investigated under this framework and a link to the theory of modulus semigroups and monotone semiflows is also discussed.
... For specific types of equations, namely delay differential equations with discrete delays, the bifurcation analysis can be performed with numerical packages like, for instance, dde-biftool [24] , knut [41] or xpp [25] . For specific models of physiologically structured populations of the type described above, some free software allows to perform the numerical analysis of equilibria, including their continuation with respect to one or two parameters, the computation of stability, and the analysis of evolutionary dynamics [6,16] . Moreover, some matlab codes are available for the stability analysis of equilibria and periodic solutions of linear delay equations [7,8,10] . ...
We address the problem of the numerical bifurcation analysis of general nonlinear delay equations, including integral and integro-differential equations, for which no software is currently available. Pseudospectral discretization is applied to the abstract reformulation of equations with infinite delay to obtain a finite dimensional system of ordinary differential equations, whose properties can be numerically studied with well-developed software. We explore the applicability of the method on some test problems and provide some numerical evidence of the convergence of the approximations.
Delay equations generate dynamical systems on infinite-dimensional state spaces. Their stability analysis is not immediate and reduction to finite dimension is often the only chance. Numerical collocation via pseudospectral techniques recently emerged as an efficient solution. In this part we analyze the application of these methods to discretize the infinitesimal generator of the semigroup of solution operators associated to the system. The focus is on both local stability of equilibria and general bifurcation analysis of nonlinear problems, for either delay differential and renewal equations.
Stability and bifurcation analyses of delay equations represent fundamental challenges in many applications, emerging in important fields like, e.g., control theory and population dynamics. Exact approaches are usually unattainable due to the infinite dimension of the associated state space, so that resorting to numerical methods is unavoidable. This work is a survey on the use of collocation techniques to address the problems above from the numerical standpoint. As such it summarizes the main contributions in this context of the research of the authors in the last 15 years or so. Methods for the linearized stability analysis of equilibria and periodic solutions, as well as for the bifurcation analysis of general nonlinear systems are illustrated, together with an essential overview of relevant problems and potential developments.
The dynamics of a specific consumer-resource model for Daphnia magna is studied from a numerical point of view. In this study, Malthusian, chemostatic, and Gompertz growth laws for the evolution of the resource population are considered, and the resulting global dynamics of the model are compared as different parameters involved in the model change. In the case of Gompertz growth law, a new complex dynamic is found as the carrying capacity for the resource population increases. The numerical study is carried out with a second-order scheme that approximates the size-dependent density function for individuals in the consumer population. The numerical method is well adapted to the situation in which the growth rate for the consumer individuals is allowed to change the sign and, therefore, individuals in the consumer population can shrink in size as time evolves. The numerical simulations confirm that the shortage of the resource has, as a biological consequence, the effective shrink in size of individuals of the consumer population. Moreover, the choice of the growth law for the resource population can be selected by how the dynamics of the populations match with the qualitative behaviour of the data.
Gene Regulatory Matrices (GRMs) are a well known technique for modelling the interactions between genes. This technique is used here, but with genes and hormones, to create Gene and Hormone Regulatory Matrices (GHRMs). In addition, a network (a directed weighted graph) is constructed from the underlying interactions of several mRNA encoding enzymes and receptors and two hormones: estradiol (E2) and progesterone (P4). This also permits comparison of the impact of each given environmental condition on E2 and P4 production, as well as mRNA expression levels. Apart from differential equations techniques (which require knowledge of rates of decay of a given hormone and mRNA) there is no existing technique to accurately predict the concentration of hormones based on the concentration of mRNA. This novel approach using GHRMs permits the use of nodes to accurately model the concentrations of the remaining ones. Experiments were performed to collect data on the gene expression and hormone concentration levels for primary bovine granulosa cells under different treatments. This data was used to build the GHRM models.
Collocation methods can be applied in different ways to delay models, e.g., to detect stability of equilibria, Hopf bifurcations and compute periodic solutions to name a few. On the one hand, piecewise polynomials can be used to approximate a periodic solution for some fixed values of the model parameters, possibly using an adaptive mesh. On the other hand, polynomial collocation can be used to reduce delay systems to systems of ordinary differential equations and established continuation tools are then applied to analyze stability and detect bifurcations. These techniques are particularly useful to treat realistic models describing structured populations, where delay differential equations are coupled with renewal equations and vital rates are given implicitly as solutions of external equations, which in turn change with model parameters. In this work we show how collocation can be used to improve the performance of continuation for such complex models and to compute periodic solutions of coupled problems.
Recently, systems of coupled renewal and retarded functional differential equations have begun to play a central role in complex and realistic models of population dynamics. In view of studying the local asymptotic stability of equilibria and (mainly) periodic solutions, we propose a pseudospectral collocation method to approximate the eigenvalues of the evolution operators of linear coupled equations, providing rigorous error and convergence analyses and numerical tests. The method combines the ideas of the analogous techniques developed separately for renewal equations and for retarded functional differential equations. Coupling them is not trivial, due to the different state spaces of the two classes of equations, as well as to their different regularization properties.
A numerical method based on pseudospectral collocation is proposed to approximate the eigenvalues of evolution operators for linear renewal equations, which are retarded functional equations of Volterra type. Rigorous error and convergence analyses are provided, together with numerical tests. The outcome is an efficient and reliable tool which can be used, for instance, to study the local asymptotic stability of equilibria and periodic solutions of nonlinear autonomous renewal equations. Fundamental applications can be found in population dynamics, where renewal equations play a central role.
With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer–resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class, we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally, we apply the methods to the above models using existing data for Daphnia magna consuming Algae and for Perca fluviatilis feeding on Daphnia magna. In particular, we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.
For state-dependent delay equations, it may easily happen that the equation is not differentiable. This hampers the formulation and the pr∞f of the Principle of Linearized Stability. The fact that an equation is not dif-ferentiable does not, by itself, imply that the solution operators are not dif-ferentiable. And indeed, the aim of this paper is to present a simple example with differentiable solution operators despite of lack of differentiability of the equation. The example takes the form of a renewal equation and is motivated by a population dynamical model.
We are interested in the asymptotic stability of equilibria of structured populations modelled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on studying the characteristic equation of the linearized system is often involved or even unattainable. Therefore, we propose and investigate a numerical method to compute the eigenvalues of the associated infinitesimal generator. The latter is discretized by using a pseudospectral approach, and the eigenvalues of the resulting matrix are the sought approximations. An algorithm is presented to explicitly construct the matrix from the model coefficients and parameters. The method is tested first on academic examples, showing its suitability also for a class of mathematical models much larger than that mentioned above, including neutral- and mixed-type equations. Applications to cannibalism and consumer–resource models are then provided in order to illustrate the efficacy of the proposed technique, especially for studying bifurcations.
We investigate the properties of an (age, size) -structured model for a population of Daphnia that feeds on a dynamical algal food source. The stability of the internal equilibrium is studied in detail and combined
with numerical studies on the dynamics of the model to obtain insight in the relation between individual behaviour and population
dynamical phenomena. Particularly the change in the (age, size)-relation with a change in the food availability seems to be
an important behavioural mechanism that strongly influences the dynamics. This influence is partly stabilizing and partly
destabilizing and leads to the coexistence of a stable equilibrium and a stable limit cycle or even coexistence of two stable
limit cycles for the same parameter values. The oscillations in this case are characterized by drastic changes in the size-structure
of the population during a cycle. In addition the model exhibits the usual predator-prey oscillations that characterize Lotka-Volterra
models.
In [D. Breda, S. Maset, and R. Vermiglio, IMA J. Numer. Anal., 24 (2004), pp. 1--19.] and [D. Breda, The Infinitesimal Generator Approach for the Computation of Characteristic Roots for Delay Differential Equations Using BDF Methods, Research report UDMI RR17/2002, Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Udine, Italy, 2002.] the authors proposed to compute the characteristic roots of delay differential equations (DDEs) with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by Runge--Kutta (RK) and linear multistep (LMS) methods, respectively. In this work the same approach is proposed in a new version based on pseudospectral differencing techniques. We prove the "spectral accuracy" convergence behavior typical of pseudospectral schemes, as also illustrated by some numerical experiments.
We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023-1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman-Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254-274, 1984; de Roos et al. in J Math Biol 28:609-643, 1990) and a model introduced by Gurney-Nisbet (Theor Popul Biol 28:150-180, 1985) and Jones et al. (J Math Anal Appl 135:354-368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.
In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for "Daphnia consuming algae" models in C-code. The results obtained by way of this implementation are shown in the form of graphs.
1 Introduction.-The Simplest Delay Equation.-Delayed Negative Feedback: A Warm-Up.- Existence of Solutions.- Linear Systems and Linearization.- Semidynamical Systems and Delay Equations.- Hopf Bifurcation.- Distributed Delay Equations and the Linear Chain Trick.- Phage and Bacteria in a Chemostat.-References.- Index.
The subject of this paper is the analysis of the equibria of a SIR type epidemic model, which is taken as a case study among the wide family of dynamical systems of infinite dimension. For this class of systems both the determination of the stationary solutions and the analysis of their local asymptotic stability are often unattainable theoretically, thus requiring the application of existing numerical tools and/or the development of new ones. Therefore, rather than devoting our attention to the SIR model’s features, its biological and physical interpretation or its theoretical mathematical analysis, the main purpose here is to discuss how to study its equilibria numerically, especially as far as their stability is concerned. To this end, we briefly analyze the construction and solution of the system of nonlinear algebraic equations leading to the stationary solutions, and then concentrate on two numerical recipes for approximating the stability determining values known as the characteristic roots. An algorithm for the purpose is given in full detail. Two applications are presented and discussed in order to show the kind of results that can be obtained with these tools.
We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw-Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log((z+1)/(z-1)) in the complex plane. Gauss quadrature corresponds to Padé approximation at z = ∞. Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at z = ∞ is only half as high, but which is nevertheless equally accurate near [-1, 1].
The eigenvalues of Chebyshev and Legendre spectral differentiation matrices, which determine the allowable time step in an explicit time integration, are extraordinarily sensitive to rounding errors and other perturbations. On a grid of N points per space dimension, machine rounding leads to errors in the eigenvalues of size . This phenomenon may lead to inconsistency between predicted and observed time step restrictions. One consequence of it is that spectral differentiation by interpolation in Legendre points, which has a favorable time step restriction for the model problem in theory, is subject to an restriction in practice. The same effect occurs with Chebyshev points for the model problem . Another consequence is that a spectral calculation with a fixed time step may be stable in double precision but unstable in single precision. We know of no other examples in numerical computation of this kind of precision-dependent stability.
By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.
A family of embedded Runge-Kutta formulae RK5 (4) are derived. From these are presented formulae which have (a) ‘small’ principal truncation terms in the fifth order and (b) extended regions of absolute stability.
In this paper we give integral expressions for the elements of the inverses of second-order pseudospectral differentiation matrices. Simple upper bounds are given for the maximum norms of these inverse matrices when Chebyshev collocation points are used. Comment is made on the failure to obtain upper bounds that are uniform in the number of collocation points when the points are evenly spaced. We also give integral expressions for inverses of first-order Chebyshev pseudospectral differentiation matrices.
We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new.