scholarly journals Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

2022 ◽  
Vol 47 (1) ◽  
pp. 305-324
Author(s):  
Claudia Anedda ◽  
Fabrizio Cuccu
Keyword(s):  
Growth Rate ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Spatial Arrangement ◽  
Optimal Choice ◽  
Dirichlet Boundary Conditions ◽  
General Context ◽  
Heterogeneous Environments ◽  
Local Growth ◽  
Total Size

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem \(-\Delta u =\lambda m u \) in a bounded smooth domain \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), under homogeneous Dirichlet boundary conditions, where \(\lambda \in \mathbb{R}\) and \(m\in L^\infty(\Omega)\). The domain \(\Omega\) represents the environment and \(m(x)\), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights \(m(x)\) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive. In our work we consider a sort of refinement of the result above. We write the weight \(m(x)\) as sum of two (or more) terms, i.e. \(m(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) and \(f_2(x)\) represent the spatial densities of the two resources which contribute to form the local growth rate \(m(x)\). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of \(f_1(x)\) and \(f_2(x)\) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if \(\Omega\) is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions.

Diffusive logistic equations with indefinite weights: population models in disrupted environments

1989 ◽  
Vol 112 (3-4) ◽  
pp. 293-318 ◽  
Author(s):  
Robert Stephen Cantrell ◽  
Chris Cosner
Keyword(s):  
Growth Rate ◽  
Diffusion Rate ◽  
Spatial Arrangement ◽  
Lower Solution ◽  
Continuation Methods ◽  
Singular Perturbation Theory ◽  
Local Growth ◽  
Positive Part ◽  
Logistic Equations ◽  
Indefinite Weights

SynopsisThe dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form ut = d Δu + [m(x) — cu]u in Ω × (0, ∞), where u represents the population density, c, d > 0 are constants describing the limiting effects of crowding and the diffusion rate of the population, respectively, and m(x) describes the local growth rate of the population. If the environment ∞ is bounded and is surrounded by uninhabitable regions, then u = 0 on ∂∞× (0, ∞). The growth rate m(x) is positive on favourablehabitats and negative on unfavourable ones. The object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled. The models are shown to possess a unique, stable, positive steady state (implying persistence for the population) provided l/d> where is the principle positive eigenvalue for the problem — Δϕ=λm(x)ϕ in Χ,ϕ=0 on ∂Ω. Analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat. In the limit as the diffusion rate d ↓ 0, the solutions tend toward the positive part of m(x)/c, and if m is discontinuous develop interior transition layers. The analysis uses bifurcation and continuation methods, the variational characterisation of eigenvalues, upper and lower solution techniques, and singular perturbation theory.

scholarly journals Multiple solutions of fourth-order difference equations with different boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yuhua Long ◽  
Shaohong Wang ◽  
Jiali Chen
Keyword(s):  
Boundary Conditions ◽  
Fourth Order ◽  
Difference Equations ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Dirichlet Boundary Conditions ◽  
Invariant Sets ◽  
Mountain Pass Lemma ◽  
Order Difference

Abstract In the present paper, a class of fourth-order nonlinear difference equations with Dirichlet boundary conditions or periodic boundary conditions are considered. Based on the invariant sets of descending flow in combination with the mountain pass lemma, we establish a series of sufficient conditions on the existence of multiple solutions for these boundary value problems. In addition, some examples are provided to demonstrate the applicability of our results.

Dynamic behavior of a stochastic SIRS model with two viruses

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiandong Zhao ◽  
Tonghua Zhang ◽  
Zhixia Han
Keyword(s):  
Growth Rate ◽  
Asymptotic Stability ◽  
Global Existence ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Environmental Noise ◽  
Sirs Model ◽  
The Moment ◽  
Disease Free

AbstractTo study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.

scholarly journals Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability

2014 ◽  
Vol 51 (4) ◽  
pp. 1114-1132 ◽  
Author(s):  
Bernhard C. Geiger ◽  
Christoph Temmel
Keyword(s):  
Growth Rate ◽  
Markov Chain ◽  
Sufficient Conditions ◽  
Entropy Rate ◽  
Transition Graph ◽  
Order Markov Chain ◽  
Finite State ◽  
Random Growth ◽  
Irreducible Markov Chain ◽  
Finite State Space

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

scholarly journals Fast and slow points of Birkhoff sums

2019 ◽  
Vol 40 (12) ◽  
pp. 3236-3256
Author(s):  
FRÉDÉRIC BAYART ◽  
ZOLTÁN BUCZOLICH ◽  
YANICK HEURTEAUX
Keyword(s):  
Growth Rate ◽  
Continuous Function ◽  
Unit Circle ◽  
General Context ◽  
Typical Element ◽  
Typical Part

We investigate the growth rate of the Birkhoff sums $S_{n,\unicode[STIX]{x1D6FC}}f(x)=\sum _{k=0}^{n-1}f(x+k\unicode[STIX]{x1D6FC})$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb{T}$ and $(\unicode[STIX]{x1D6FC},x)$ is a ‘typical’ element of $\mathbb{T}^{2}$. The answer depends on the meaning given to the word ‘typical’. Part of the work will be done in a more general context.

GENETIC LOADS IN HETEROGENEOUS ENVIRONMENTS

Genetics ◽  
1975 ◽  
Vol 80 (3) ◽  
pp. 621-635
Author(s):  
Charles E Taylor
Keyword(s):  
Population Structure ◽  
Genetic Variation ◽  
Gene Frequency ◽  
Phase Plane ◽  
Genetic Load ◽  
Sufficient Conditions ◽  
Heterogeneous Environments ◽  
Niche Space ◽  
Necessary And Sufficient ◽  
Existence Of Equilibria

ABSTRACT A model of population structure in heterogeneous environments is described with attention focused on genetic variation at a single locus. The existence of equilibria at which there is no genetic load is examined.—The absolute fitness of any genotype is regarded as a function of location in the niche space and the population density at that location. It is assumed that each organism chooses to live in that habitat in which it is most fit ("optimal habitat selection").—Equilibria at which there is no segregational load ("loadless equilibria") may exist. Necessary and sufficient conditions for the existence of such equilibria are very weak. If there is a sufficient amount of dominance or area in which the alleles are selectively neutral, then there exist equilibria without segregational loads. In the N, p phase plane defined by population size, N, and gene frequency, p, these equilibria generally consist of a line segment which is parallel to the p axis. These equilibria are frequently stable.

scholarly journals Selection and spatial arrangement of building materials during the construction of nest turrets by grass-cutting ants

2020 ◽  
Vol 7 (10) ◽  
pp. 201312
Author(s):  
Daniela Römer ◽  
Marcela I. Cosarinsky ◽  
Flavio Roces
Keyword(s):  
Building Materials ◽  
Material Selection ◽  
Spatial Location ◽  
Spatial Arrangement ◽  
The Self ◽  
Nest Entrance ◽  
Self Organized ◽  
The Impact ◽  
Over Time ◽  
Selection Of

Ants build complex nest structures by reacting to simple, local stimuli. While underground nests result from the space generated by digging, some leaf- and grass-cutting ants also construct conspicuous aboveground turrets around nest openings. We investigated whether the selection of specific building materials occurs during turret construction in Acromyrmex fracticornis grass-cutting ants, and asked whether single building decisions at the beginning can modify the final turret architecture. To quantify workers' material selection, the original nest turret was removed and a choice between two artificial building materials, thin and thick sticks, was offered for rebuilding. Workers preferred thick sticks at the very beginning of turret construction, showed varying preferences thereafter, and changed to prefer thin sticks for the upper, final part of the turret, indicating that they selected different building materials over time to create a stable structure. The impact of a single building choice on turret architecture was evaluated by placing artificial beams that divided a colony's nest entrance at the beginning of turret rebuilding. Splitting the nest entrance led to the self-organized construction of turrets with branched galleries ending in multiple openings, showing that the spatial location of a single building material can strongly influence turret morphology.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

2019 ◽  
Vol 29 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Zonghong Feng ◽  
Xinxing Wu ◽  
Luo Yang
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Drug Therapy ◽  
Tumor Growth ◽  
Tumor Cells ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Tumor Growth Rate ◽  
Chaotic Region ◽  
Treatment Parameter

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].

scholarly journals Radial Growth Patterns Associated with Tree Mortality in Nothofagus pumilio Forest

Forests ◽  
2019 ◽  
Vol 10 (6) ◽  
pp. 489 ◽  
Author(s):  
Milagros Rodríguez-Catón ◽  
Ricardo Villalba ◽  
Ana Srur ◽  
A. Park Williams
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Forest Dynamics ◽  
Radial Growth ◽  
Tree Mortality ◽  
Growth Patterns ◽  
Severe Drought ◽  
Nothofagus Pumilio ◽  
Total Size ◽  
Growth Trends

Tree mortality is a key process in forest dynamics. Despite decades of effort to understand this process, many uncertainties remain. South American broadleaf species are particularly under-represented in global studies on mortality and forest dynamics. We sampled monospecific broadleaf Nothofagus pumilio forests in northern Patagonia to predict tree mortality based on stem growth. Live or dead conditions in N. pumilio trees can be predicted with high accuracy using growth rate as an explanatory variable in logistic models. In Paso Córdova (CO), Argentina, where the models were calibrated, the probability of death was a strong negative function of radial growth, particularly during the six years prior to death. In addition, negative growth trends during 30 to 45 years prior to death increased the accuracy of the models. The CO site was affected by an extreme drought during the summer 1978–1979, triggering negative trends in radial growth of many trees. Individuals showing below-average and persistent negative trends in radial growth are more likely to die than those showing high growth rates and positive growth trends in recent decades, indicating the key role of droughts in inducing mortality. The models calibrated at the CO site showed high verification skill by accurately predicting tree mortality at two independent sites 76 and 141 km away. Models based on relative growth rates showed the highest and most balanced accuracy for both live and dead individuals. Thus, the death of individuals across different N. pumilio sites was largely determined by the growth rate relative to the total size of the individuals. Our findings highlight episodic severe drought as a triggering mechanism for growth decline and eventual death for N. pumilio, similar to results found previously for several other species around the globe. In the coming decades, many forests globally will be exposed to more frequent and/or severe episodes of reduced warm-season soil moisture. Tree-ring studies such as this one can aid prediction of future changes in forest productivity, mortality, and composition.

scholarly journals Accelerate stochastic subgradient method by leveraging local growth condition

2019 ◽  
Vol 17 (05) ◽  
pp. 773-818 ◽  
Author(s):  
Yi Xu ◽  
Qihang Lin ◽  
Tianbao Yang
Keyword(s):  
Growth Rate ◽  
Global Convergence ◽  
Objective Function ◽  
Optimal Solution ◽  
Subgradient Method ◽  
Local Region ◽  
Subgradient Methods ◽  
Local Growth ◽  
Smoothness Assumption ◽  
First Order

In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function [Formula: see text] in the [Formula: see text]-sublevel set grows as fast as [Formula: see text], where [Formula: see text] represents the closest optimal solution to [Formula: see text] and [Formula: see text] quantifies the local growth rate, the iteration complexity of first-order stochastic optimization for achieving an [Formula: see text]-optimal solution can be [Formula: see text], which is optimal at most up to a logarithmic factor. To achieve the faster global convergence, we develop two different accelerated stochastic subgradient methods by iteratively solving the original problem approximately in a local region around a historical solution with the size of the local region gradually decreasing as the solution approaches the optimal set. Besides the theoretical improvements, this work also includes new contributions toward making the proposed algorithms practical: (i) we present practical variants of accelerated stochastic subgradient methods that can run without the knowledge of multiplicative growth constant and even the growth rate [Formula: see text]; (ii) we consider a broad family of problems in machine learning to demonstrate that the proposed algorithms enjoy faster convergence than traditional stochastic subgradient method. We also characterize the complexity of the proposed algorithms for ensuring the gradient is small without the smoothness assumption.

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