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A Model of Physiological Structured Population Dynamics with a Nonlinear Individual Growth Rate
Abstract
In this article we consider a size structured population model with a nonlinear growth rate depending on the individual's size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.
... Models of age-structured populations using partial differential equations, such as those originally discribed by A. Lotka, A.G. McKendrick, W.O. Kermack, and F. von Foerster, are well adapted to model the dynamical features of experimental cultures of cells transiting the cell cycle with variable IMT. These models have been widely studied from a mathematical perspective [4,3,15,16,17,22,37,40,43,54], but the application of the model to experimental data has been hampered by an inability to determine the age-dependent model parameters. The usual approach for parameter estimation is to solve numerically an inverse problem (see [1,9,10,20,19,23,28,42,44,45,51,52,41] on this question for structured population models), but this requires extensive input data and is specific to a given situation. ...
... The different functions (15) to (18) It has been observed that an exponentially modified Gaussian (EMG) is often a better model for IMT distributions than the gamma function [27,53] . An EMG is defined as the convolution of a Gaussian with a decreasing exponential, but after solving it can be written with three parameters as ...
... Step 2: Obtain parameters from model fit to IMT distribution a) Choose a form for I ∞ as a gamma function (15) or (17), or as the new EMG form (21). b) Fit the histogram (H i ) with the corresponding formĨ ∞ from definition (25). For PC-9 cancer cells we choose the form (21), because in [53] it was observed that the IMT distribution appeared to be an EMG (see Figure 5 for the example). ...
Cells grown in culture act as a model system for analyzing the effects of anticancer compounds, which may affect cell behavior in a cell cycle position-dependent manner. Cell synchronization techniques have been generally employed to minimize the variation in cell cycle position. However, synchronization techniques are cumbersome and imprecise and the agents used to synchronize the cells potentially have other unknown effects on the cells. An alternative approach is to determine the age structure in the population and account for the cell cycle positional effects post hoc. Here we provide a formalism to use quantifiable age distributions from live cell microscopy experiments to parameterize an age-structured model of cell population response.
... On the other hand, the dynamics of coagulating particles has been investigated using similar equations as in (1.2) [1]. Existence theory have been established using the characteristic method with the contraction mapping principle [6], the semigroup theorem [4] or the upper-lower solution method [3]. ...
... Using arguments in [6], we can establish P ( u(·, t)) is continuous. We claim u is unique. ...
In this paper, we study a free boundary problem for a class of nonlinear nonautonomous size structured population model. Using the comparison principle and upper lower solution methods, we establish the existence of the solution for such kind of a model.
... After that, Chipot [10] improved the result by using a fixed-point argument. See also [8,30,32,34,35] for physiologically structured population models. ...
... δ(a)U (t, a)φ ′ (a)da − µ(S, P ; t, a)U (t, a)φ(a)da in D ′ (0, T ). Moreover, U (t, a) defined in (9) is a weak solution in L ∞ (0, T ; L 1 (R + )) ∩ L ∞ ([0, T ] × R + )of (7) satisfying the initial and boundary conditions in(8). ...
... Now we should address the issue of feedback: how is the environmental condition (partly) determined by interaction with the focal population? For an attempt at building a general framework see Diekmann et al. (2001), and for particular examples see Calsina and Saldaña (1995), Barril et al. (2022), and Clément et al. (2024). The general idea is to formulate a fixed point problem and to show that the contraction mapping principle can be applied, to obtain a unique solution (probably first on a small time interval, but by continuation on a maximal time interval). ...
This chapter focuses on variable maturation delay or, more precisely, on the mathematical description of a size-structured population consuming an unstructured resource. When the resource concentration is a known function of time, we can describe the growth and survival of individuals quasi-explicitly, i.e., in terms of solutions of ordinary differential equations (ODE). Reproduction is captured by a (non-autonomous) renewal equation, which can be solved by generation expansion. After these preparatory steps, a contraction mapping argument is needed to construct the solution of the coupled consumer-resource system with prescribed initial conditions. As we shall show, this interpretation-guided constructive approach does in fact yield weak solutions of a familiar partial differential equation (PDE). A striking difficulty with the PDE approach is that the solution operators are, in general, not differentiable, precluding a linearized stability analysis of steady states. This is a manifestation of the state-dependent delay difficulty. As a (not entirely satisfactory and rather technical) way out, we present a delay equation description in terms of the history of both the p-level birth rate of the consumer population and the resource concentration. We end by using pseudospectral approximation to derive a system of ODE and demonstrating its use in a numerical bifurcation analysis. Importantly, the state-dependent delay difficulty dissolves in this approximation.
... Inspired by [4], we use the characteristic line method to simplify the original equation into a pair of coupled integral equations. ...
In this paper, the optimal control problem of a forest population system with size structure in a polluted environment is established and studied by considering the influence of competition within the forest population on the size growth rate. The solution of the system is obtained by the characteristic line method, and the existence and uniqueness of the solution are proved by the fixed point theorem. The existence of the optimal control strategy is proved by the maximum sequence after introducing the separable model, and then the necessary conditions of the optimal control are obtained by means of Pontryagin’s maximum principle and Hamiltonian function.
... Several mathematicians then studied these problems, including; Huyer [15] for a size-structured population, and Maltus' model in 1798, which is based on the number of individuals and calculates the quantitative evolution of the population with a rate of increase. A number of mathematicians carried out a rigorous analysis of these models, in order to improve them, its mathematicians include; Feller [16] in 1941, the model of; Sinko and Streifer [17,18] which is a model used in the study of populations structured by age and size, and the work of; Chan and Guo [19], which is a model used in the study of populations structured by age and size, we can also see the work of Rees and Ellner [20] which is an age-structured model. ...
We consider a linear system based on a population dynamics model dependent on age, space and nonlocal boundary conditions. Note that our population dynamics model is a four-step model with a second derivative with respect to the age variable and a second derivative with respect to the space variable. Population growth at each stage depends on time and space. We prove the uniqueness and existence of a positive solution by combining the Galerkin’s variational method and the Banach’s fixed point theorem.
... The existence and uniqueness of solutions for linear and nonlinear size-structured population models have been extensively investigated through various methods in the literature. In addition to the upper and lower solution approach mentioned earlier, other traditional methodologies include the characteristic method with fixed point argument (e.g., (Calsina and Saldana 1995)) and the method of the semigroups of linear operators (e.g., (Banks and Kappel 1989;Banks et al. 1994;Pazy 1983)). However, the general semigroup theory on semi-linear PDE in Pazy (1983) might not be applicable to our model due to the non-linearity in the boundary condition, specifically the dependence of the reproduction rate function on the total population. ...
In this paper, we propose and analyze a nonautonomous model that describes the dynamics of a size-structured consumer interacting with an unstructured resource. We prove the existence and uniqueness of the solution of the model using the monotone method based on a comparison principle. We derive conditions on the model parameters that result in persistence and extinction of the population via the upper-lower solution technique. We verify and complement the theoretical results through numerical simulations.
... that are cut per unit of time (10 years) per ha is the wood volume extraction rate and it is considered a surrogate of the forest yield, denoted by w(t): System (7)-(9 possesses a unique solution under simple hypotheses on functions g(x, V) , m(x) and u 0 (x) (Calsina and Saldaña 1995;Kato 2004). This solution, the temporal evolution of the tree distribution, has not an explicit expression but it is approximated as sharply as needed by the appropriate numerical scheme. ...
Context
Mediterranean managed dry-edge pine forests maintain biodiversity and supply key ecosystem services but are threatened by climate change and are highly vulnerable to desertification. Forest management through its effect on stand structure can play a key role on forest stability in response to increasing aridity, but the role of forest structure on drought resilience remains little explored.
Objectives
To investigate the role of tree growth and forest structure on forest resilience under increasing aridity and two contrasting policy-management regimes. We compared three management scenarios; (i) “business as usual”-based on the current harvesting regime and increasing aridity—and two scenarios that differ in the target forest function; (ii) a “conservation scenario”, oriented to preserve forest stock under increasing aridity; and (iii), a “productivity scenario” oriented to maintain forest yield under increasingly arid conditions.
Methods
The study site is part of a large-homogeneous pine-covered landscape covering sandy flatlands in Central Spain. The site is a dry-edge forest characterized by a lower productivity and tree density relative to most Iberian Pinus pinaster forests. We parameterized and tested an analytical size-structured forest dynamics model with last century tree growth and forest structure historical management records.
Results
Under current management (Scenario-i), increasing aridity resulted in a reduction of stock, productivity, and maximum mean tree size. Resilience boundaries differed among Scenario-ii and -Scenario-iii, revealing a strong control of the management regime on resilience via forest structure. We identified a trade-off between tree harvest size and harvesting rate, along which there were various possible resilient forest structures and management regimes. Resilience boundaries for a yield-oriented management (Scenario-iii) were much more restrictive than for a stock-oriented management (Scenario-ii), requiring a drastic decrease in both tree harvest size and thinning rates. In contrast, stock preservation was feasible under moderate thinning rates and a moderate reduction in tree harvest size.
Conclusions
Forest structure is a key component of forest resilience to drought. Adequate forest management can play a key role in reducing forest vulnerability while ensuring a long-term sustainable resource supply. Analytical tractable models of forest dynamics can help to identify key mechanisms underlying drought resilience and to design management options that preclude these social-ecological systems from crossing a tipping point over a degraded alternate state.
A solution is found by means of a population of structured sizes through the method of characteristics lines. The evolution of the population is represented by linear partial differential equations of first order. The population model considered is used to show the impact of the distribution initial sizes in culture aquatics systems.
The asymptotic dynamics of a system of ordinary differential equations describing the dynamics of nsize-structured species competing for a single (unstructured) resource are studied. The system is based on a single species growth model for a size-structured species due to Diekmann, Metz, Kooijman, and Heijmans in which physiological parameters at the level of the individual are incorporated. It is shown that all trajectories asymptotically approach a lower-dimensional positive cone where the dynamics are governed by an easily determined lower-dimensional competition system of a type commonly studied in the literature for unstructured populations. It is also shown that, regardless of the asymptotic dynamics or the outcome of the competitive interaction, the average size of individuals for every species asymptotically equilibrates to a positive value. These results permit a study of competitive exclusion in terms of the physiological parameters and average size of individuals of the species. Illustrative ap...
We heuristically discuss the well-posedness of three variants of the S. A. L. M. Kooijman and J. A. J. Metz model [Ecotox. 8, 254-274 (1984)]. Shortcomings concerning the uniqueness and continuous dependence on data of the solutions to one of the variants are traced back to an inconsistency in the biological concept of energy allocation in this model version. The conceptional consequences are discussed and an open question concerning energy allocation is pin-pointed.
A nonlinear model is presented for the dynamics of a population in which each individual is characterized by its chronological age and by an arbitrary finite number of additional structure variables. The nonlinearities are introduced by assuming that the birth and loss processes, as well as the maturation rates of individuals, are controlled by a functional of the population density. The model is a generalization of the classical Sharpe-Lotka-McKendrick model of age-structured population growth, the nonlinear age-structured model of Gurtin and MacCamy, and the age-size-structured cell population model of Bell and Anderson. Based on a reformulation of the model in terms of a coupled system of equations, the existence for all positive time of unique solutions to the model is proved using a contraction mapping argument. The existence of equilibrium solutions is discussed, and sufficient conditions are proved for the local asymptotic stability of equilibria using results from the theory of strongly continuous...
An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.
The concept of persistence reflects the survival of all components of a model ecosystem. Most of the results to date are restricted to ordinary differential equations or to dynamics on locally compact spaces. The concept is investigated here in the setting of a -semigroup which is asymptotically smooth. Since the equations of population dynamics often involve delays or diffusion this seems the appropriate setting. Conditions are placed on the flow on the boundary which, given the presence of a global attractor provided by the assumption of dissipativeness and asymptotic smoothness, are necessary and sufficient for persistence.