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  • 1. Banerjee, M.
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stabilizing Role of Nonlocal Interaction on Spatio-temporal Pattern Formation2016In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 11, no 5, p. 103-118Article in journal (Refereed)
    Abstract [en]

    Here we study a spatio-temporal prey-predator model with ratio-dependent functional response and nonlocal interaction term in the prey growth. For a clear understanding of the effect of nonlocal interaction on the resulting stationary and non-stationary patterns, we consider the nonlocal interaction term in prey growth only to describe the nonlocal intra-specific competition due to limited resources for the prey. First we obtain the patterns exhibited by the basic model in the absence of nonlocal interaction and then explore the effect of nonlocal interaction on the resulting patterns. We demonstrate the stabilizing role of nonlocal interaction as it induces stationary pattern from periodic and chaotic regimes with an increase in the range of nonlocal interaction. The existence of multiple branches of stationary solutions, bifurcating from homogeneous steady-state as well as non-stationary patterns, is illustrated with the help of numerical continuation technique.

  • 2. Banerjee, Malay
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Influence of discrete delay on pattern formation in a ratio-dependent prey-predator model2014In: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 67, p. 73-81Article in journal (Refereed)
    Abstract [en]

    In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large. 

  • 3. Banerjee, Malay
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Time delay can enhance spatio-temporal chaos in a prey-predator model2016In: Ecological Complexity: An International Journal on Biocomplexity in the Environment and Theoretical Ecology, ISSN 1476-945X, E-ISSN 1476-9840, Vol. 27, p. 17-28Article in journal (Refereed)
    Abstract [en]

    In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey-predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns. 

  • 4.
    Cao, Zhanghua
    et al.
    Nantong Univ, Sch Elect & Informat, Nantong 226019, Jiangsu, Peoples R China. .
    Zhang, Shibing
    Nantong Univ, Sch Elect & Informat, Nantong 226019, Jiangsu, Peoples R China. .
    Ji, Xiaodong
    Nantong Univ, Sch Elect & Informat, Nantong 226019, Jiangsu, Peoples R China. .
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Secure random linear network coding on a wiretap network2015In: AEU - International Journal of Electronics and Communications, ISSN 1434-8411, E-ISSN 1618-0399, Vol. 69, no 1, p. 467-472Article in journal (Refereed)
    Abstract [en]

    We develop a secure random linear network coding scheme on wiretap networks where a wiretapper can only eavesdrop on a limited number of channels. On one hand, by refining Lima's "locked coefficients" method and applying the approach of one-time pad, our scheme can well protect message packets without decreasing network throughput. On the other hand, by treating ciphertext as noisy symbols, inspired by the physical layer technique, and applying Shamir's secret sharing scheme, our scheme can successfully protect secret random seed without any forms of key exchange or secret channels. Compared to existing schemes, our scheme has minimum information overhead, independency of hash functions, and no restriction on global encoding kernel. Finally, we analyze the computational complexity of our proposed scheme and rigorously prove that our scheme can achieve secure network communication. (C) 2014 Elsevier GmbH. All rights reserved.

  • 5. Ge, Jing
    et al.
    Lin, Ling
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A DIFFUSIVE SIS EPIDEMIC MODEL INCORPORATING THE MEDIA COVERAGE IMPACT IN THE HETEROGENEOUS ENVIRONMENT2017In: Discrete and continuous dynamical systems. Series B, ISSN 1531-3492, E-ISSN 1553-524X, Vol. 22, no 7, p. 2763-2776Article in journal (Refereed)
    Abstract [en]

    To explore the impact of media coverage and spatial heterogeneity of environment on the prevention and control of infectious diseases, a spatial-temporal SIS reaction-diffusion model with the nonlinear contact transmission rate is proposed. The nonlinear contact transmission rate is spatially dependent and introduced to describe the impact of media coverage on the transmission dynamics of disease. The basic reproduction number associated with the disease in the heterogeneous environment is established. Our results show that the degree of mass media attention plays an important role in preventing the spreading of infectious diseases. Numerical simulations further confirm our analytical findings.

  • 6.
    Lindh, Magnus
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Falster, Daniel
    Macquarie University, Sydney, Australia.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Dieckmann, Ulf
    Evolution and Ecology Program, International Institute for Applied Systems Analysis, Schlossplatz 1, A–2361 Laxenburg, Austria.
    Brännström, Åke
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Evolution of tree crown shape and the influence of productivity, incident sun angle, and latitude2016Manuscript (preprint) (Other academic)
    Abstract [en]

    Across the globe, large variations are observed in plant architecture, from bushes in tundra and semi desert, to high top-heavy trees in boreal and tropical forests. Despite recent advances in large scale monitoring of forests, little is known about how crown architecture varies with environmental conditions. We investigate how shading from the plant on itself, and the shading from the forest influence growth, using a dynamic size-structured crown architecture model with mean-field shading and self-shading, based on an established model. We evolve the two traits crown top-heaviness and crown width-to-height ratio.

    We report the following findings: (1) Tree crowns are shaped by trade-offs. Top-heavy crowns intercept light well as they can reach high up in the vertical light gradient, but they have low crown-rise efficiency. Wide crowns have a low leaf density per volume giving low self-shading, but a large cost for branches. (2) When coevolving the two traits we find a single evolutionarily stable strategy, far away from the strategy maximizing net primary production. (3) When only sun angle decreases with latitude, both crown width-to-height ratio and crown top-heaviness are decreasing with latitude. When both sun angle and light assimilation response to canopy openness decreases with latitude, crown width-to-height ratio is decreasing significantly only at sites with low productivity, while crown top-heaviness decreases for all sites independent of productivity. Crown top-heaviness increases with increasing site productivity, as a result of a darker forest caused by an increasing density of plants. (4) When varying latitude and sun angle over large ranges we find that crown width-to-height ratio has a maximum at intermediate net primary production or leaf area index, while crown top heaviness is saturating for high net primary production or leaf area index.

    Our model approach makes it possible to study evolving crown shapes in high detail, and we can identify trade-offs for crown shape. As expected crown top-heaviness is increasing with site productivity and net primary production, but crown width-to-height ratio has a rich and a more unexpected response due to interactions of self-shading and mean-field-shading.

  • 7.
    Lindh, Magnus
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Falster, Daniel
    Franklin, Oskar
    Brännström, Åke
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Plant diversity and drought: The role of deep roots2014In: Ecological Modelling, ISSN 0304-3800, E-ISSN 1872-7026, Vol. 290, p. 85-93Article in journal (Refereed)
    Abstract [en]

    Extreme temperatures and droughts in the wake of climate change potentially threaten plant diversity. A strategy that plants use to improve survival during seasonal drought is to establish deep roots, aptly named tap roots for their ability to tap into groundwater. Little is known, however, about the role of deep roots in maintaining plant diversity. Here, we extend an established model of plants canopies by Iwasa et al. (1985), in which plants of different heights compete for light, to allow strategic investments in an optional tap root. We investigate how emerging diversity varies with water table depth, soil water gradient and drought-induced mortality rate. Having a tap root enables plants to reach deep water, thus reducing mortality, but also carries a construction cost, thus inducing a tradeoff. We find (1) that tap roots maintain plant diversity under increasing drought mortality, (2) that tap roots evolve when ground water is accessible at low to intermediate depths, (3) no viable strategies at high drought mortality and deep water table, and (4) Red Queen evolutionary dynamics in mixed communities with and without tap root.

  • 8. Ling, Zhi
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Dynamical behavior of a harvest single species model on growing habitat2014In: Bulletin of the Korean Mathematical Society, ISSN 1015-8634, Vol. 51, no 5, p. 1357-1368Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with a reaction-diffusion single species model with harvesting on n-dimensional isotropically growing domain. The model on growing domain is derived and the corresponding comparison principle is proved. The asymptotic behavior of the solution to the Problem is obtained by using the method of upper and lower solutions. The results show that the growth of domain takes a positive effect on the asymptotic stability of positive steady state solution while it takes a negative effect on the asymptotic stability of the trivial solution, but the effect of the harvesting rate is opposite. The analytical findings are validated with the numerical simulations.

  • 9. Ling, Zhi
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Lin, Zhigui
    Turing pattern formation in a predator-prey system with cross diffusion2014In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 38, no 21-22, p. 5022-5032Article in journal (Refereed)
    Abstract [en]

    The paper explores the impacts of cross-diffusion on the formation of spatial patterns in a ratio-dependent predator-prey system with zero-flux boundary conditions. Our results show that under certain conditions, cross-diffusion can trigger the emergence of spatial patterns which is however impossible under the same conditions when cross-diffusion is absent. We give a rigorous proof that the model has at least one spatially heterogenous steady state by means of the Leray-Schauder degree theory. In addition, numerical simulations are performed to visualize the complex spatial patterns.

  • 10. Liu, Jia
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Bifurcation analysis in a prey-predator model with nonlinear predator harvesting2016In: Journal of the Franklin Institute, ISSN 0016-0032, E-ISSN 1879-2693, Vol. 353, no 17, p. 4701-4714Article in journal (Refereed)
    Abstract [en]

    In this paper, the spatiotemporal dynamics of a delayed diffusive prey predator model with nonlinear predator harvesting is investigated. Through mathematical analysis, we obtain the conditions for Turing and Hopf bifurcation. Numerical simulations display a variety of spatial patterns including spots, strips, mixture of spots and strips, spiral, patchy structure, and chaos. The delay is found to have significant influence on the emergent spatial patterns, such as changing arm length and direction of strips, and accelerating the transformation of spatial patterns. 

  • 11.
    Meng, Xinzhu
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Evolutionary dynamics in a Lotka–Volterra competition model with impulsive periodic disturbance2016In: Mathematical methods in the applied sciences, ISSN 0170-4214, E-ISSN 1099-1476, Vol. 39, no 2, p. 177-188Article in journal (Refereed)
    Abstract [en]

    In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra-type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size-selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long-term simulation of evolutionary dynamics to demonstrate the emergence of high-level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching.

  • 12.
    Portalier, Sebastien M. J.
    et al.
    Department of Biology, McGill University.
    Cherif, Mehdi
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Fussmann, Gregor F.
    Department of Biology, McGill University.
    Loreau, Michel
    Centre for Biodiversity Theory and Modelling, Station d’Ecologie Expérimentale du CNRS.
    Size-related effects of physical factors on phytoplankton communities2016In: Ecological Modelling, ISSN 0304-3800, E-ISSN 1872-7026, Vol. 323, p. 41-50Article in journal (Refereed)
    Abstract [en]

    Phytoplankton communities are influenced by light availability. Therefore, one factor promoting phytoplankton species persistence is their ability to stay within the euphotic zone. This ability is determined by the interplay between species mass, buoyancy and dispersion, which are driven by physical factors. In this study, we investigate how these physical factors and light-use efficiency, all correlated with cell size, influence species persistence. Our model shows, first, that species can persist only within a size-dependent range of turbulence strength. The minimal level of turbulence required for persistence increases drastically with cell size, while all species reach similar maximal levels of turbulence. Second, the maximal water column depth allowing persistence is also size-dependent: large cells show a maximal depth at both low and high turbulence strength, while small cells show this pattern only at high turbulence strength. This study emphasizes the importance of the physical medium in ecosystems and its interplay with cell size for phytoplankton dynamics and bloom condition.

  • 13. Tian, C.
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hopf bifurcation analysis in a diffusive food-chain model with time delay2013In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 66, no 10, p. 2139-2153Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a three species food-chain model with diffusion. By means of analyzing eigenvalue spectrum and Lyapunov functional, we show that the positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation. The instability of the Hopf bifurcation leads to the emergence of spatial patterns. Numerical calculations are performed to illustrate our theoretical results.

  • 14. Tian, Canrong
    et al.
    Lin, Ling
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Additive noise driven phase transitions in a predator-prey system2017In: Applied Mathematical Modelling, ISSN 0307-904X, E-ISSN 1872-8480, Vol. 46, p. 423-432Article in journal (Refereed)
    Abstract [en]

    We explore the impact of additive noise on phase transitions in a predator-prey system, which is formulated by stochastic partial differential equations (SPDEs). The system is observed to experience twice phase transitions under certain level of additive noise. We extend the multiple scaling approach from single SPDE to multiple SPDEs and find a necessary and sufficient condition to excite the occurrence of the first transition from a spatially homogeneous state to a spatially regular spiral wave. Numerical experiments show the second phase transition from the regular spatial pattern to spiral turbulence. We conclude that additive noise has a destabilising effect on population dynamics by triggering the onset of Hopf bifurcation.

  • 15. Tian, Canrong
    et al.
    Ling, Zhi
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nonlocal interaction driven pattern formation in a prey-predator model2017In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 308, p. 73-83Article in journal (Refereed)
    Abstract [en]

    A widely observed scenario in ecological systems is that populations interact not only with those living in the same spatial location but also with those in spatially adjacent locations, a phenomenon called nonlocal interaction. In this paper, we explore the role of nonlocal interaction in the emergence of spatial patterns in a prey-predator model under the reaction-diffusion framework, which is described by two coupled integro-differential equations. We first prove the existence and uniqueness of the global solution by means of the contraction mapping theory and then conduct stability analysis of the positive equilibrium. We find that nonlocal interaction can induce Turing bifurcation and drive the formation of stationary spatial patterns. Finally we carry out numerical simulations to demonstrate our analytical findings.

  • 16. Tian, Canrong
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Delay-driven irregular spatiotemporal patterns in a plankton system2013In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 88, no 1, p. 012713-Article in journal (Refereed)
    Abstract [en]

    An inhomogeneous distribution of species density over physical space is a widely observed scenario in plankton systems. Understanding the mechanisms resulting in these spatial patterns is a central topic in plankton ecology. In this paper we explore the impact of time delay on spatiotemporal patterns in a prey-predator plankton system. We find that time delay can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, a phase transition from a regular spiral pattern to an irregular one was observed and the latter gradually replaced the former and persisted indefinitely. The characteristic length of the emergent spatial pattern is consistent with the scale of plankton patterns observed in field studies.

  • 17.
    Tian, Canrong
    et al.
    Department of Basic Sciences, Yancheng Institute of Technology, Yancheng 224003, China .
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Traveling wave governs the stability of spatial pattern in a model of allelopathic competition interactions2012In: Chaos, ISSN 1054-1500, E-ISSN 1089-7682, Vol. 22, no 4, p. 043136-Article in journal (Refereed)
    Abstract [en]

    Inhomogenous distribution of populations across physical space is a widely observed scenario in nature and has been studied extensively. Mechanisms accounting for these observations are such as diffusion-driven instability and mechanochemical approach. While conditions have been derived from a variety of models in biological, physical, and chemical systems to trigger the emergence of spatial patterns, it remains poorly understood whether the spatial pattern possesses asymptotical stability. In a plankton allelopathic competitive system with distributed time delay, we found that spatial pattern arises as a result of Hopf bifurcation and, in the meantime, there exists a unique asymptotically stable traveling wave solution. The convergence of the traveling wave solution to the emergent pattern and its stability infer that the emergent spatial pattern is locally asymptotically stable.

  • 18. Zhang, Hong
    et al.
    Georgescu, Paul
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Periodic patterns and Pareto efficiency of state dependent impulsive controls regulating interactions between wild and transgenic mosquito populations2016In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 31, no 1-3, p. 83-107Article in journal (Refereed)
    Abstract [en]

    It is conceivable that genetically modified mosquitoes could stop the spread of malaria, by outcompeting the wild mosquitoes and interfering with their reproductive processes, and genetically inheriting and further transmitting a diminished potential to carry Plasmodium. To get insight into the possible outcomes, we formulate an ODE model for the interactions between wild and transgenic mosquito populations, which is subject to state-dependent impulsive perturbations. By first investigating the dynamics of the unperturbed system, we determine certain sufficient conditions for the existence and orbital stability of positive order-1 solution of the model system with state-dependent impulsive perturbations. Their feasibility is then illustrated by means of numerical simulations. In addition, to adequately control the wild mosquito population, we use a multi-target approach which, in addition to accounting for the total costs, keeps track of the total size of the wild mosquito population. To trade off these objectives, we consider the concept of Pareto efficiency to determine suitable control strategies which are near-optimal. Finally, we carry out numerical simulations to illustrate the Pareto frontier and then characterize the detailed Pareto efficient control regime. (C) 2015 Elsevier B.V. All rights reserved.

  • 19.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Technical University of Denmark, Denmark ; International Institute for Applied Systems Analysis, Laxenburg, Austria.
    Andersen, Ken H.
    Dieckmann, Ulf
    Brannstrom, Ake
    Four types of interference competition and their impacts on the ecology and evolution of size-structured populations and communities2015In: Journal of Theoretical Biology, ISSN 0022-5193, E-ISSN 1095-8541, Vol. 380, p. 280-290Article in journal (Refereed)
    Abstract [en]

    We investigate how four types of interference competition which alternatively affect foraging, metabolism, survival, and reproduction impact the ecology and evolution of size-structured populations. Even though all four types of interference competition reduce population biomass, interference competition at intermediate intensity sometimes significantly increases the abundance of adult individuals and the population's reproduction rate. We find that foraging and metabolic interference evolutionarily favor smaller maturation size when interference is weak and larger maturation size when interference is strong. The evolutionary response to survival interference and reproductive interference is always larger maturation size. We also investigate how the four types of interference competition impact the evolutionary dynamics and resultant diversity and trophic structure of size-structured communities. Like other types of trait-mediated competition, all four types of interference competition can induce disruptive selection and thus promote initial diversification. Even though foraging interference and reproductive interference are more potent in promoting initial diversification, they catalyze the formation of diverse communities with complex trophic structure only at high levels of interference intensity. By contrast, survival interference does so already at intermediate levels, while reproductive interference can only support relatively smaller communities with simpler trophic structure. Taken together, our results show how the type and intensity of interference competition jointly affect coexistence patterns in structured population models. (C) 2015 Elsevier Ltd. All rights reserved.

  • 20.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Dieckmann, Ulf
    Brännström, Åke
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Evolution and Ecology Program, International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria.
    On the performance of four methods for the numerical solution of ecologically realistic size-structured population models2017In: Methods in Ecology and Evolution, ISSN 2041-210X, E-ISSN 2041-210X, Vol. 8, no 8, p. 948-956Article in journal (Refereed)
    Abstract [en]

    1. Size-structured population models (SSPMs) are widely used in ecology to account for intraspecific variation in body size. Three characteristic features of size-structured populations are the dependence of life histories on the entire size distribution, intrinsic population renewal through the birth of new individuals, and the potential accumulation of individuals with similar body sizes due to determinate or stunted growth. Because of these three features, numerical methods that work well for structurally similar transport equations may fail for SSPMs and other transport-dominated models with high ecological realism, and thus their computational performance needs to be critically evaluated.

    2. Here, we compare the performance of four numerical solution schemes, the fixed-mesh upwind (FMU) method, the moving-mesh upwind (MMU) method, the characteristic method (CM), and the Escalator Boxcar Train (EBT) method, in numerically solving three reference problems that are representative of ecological systems in the animal and plant kingdoms. The MMU method is here applied for the first time to SSPMs, whereas the three other methods have been employed by other authors.

    3. Our results show that the EBT method performs best, except for one of the three reference problems, in which size-asymmetric competition affects individual growth rates. For that reference problem, the FMU method performs best, closely followed by the MMU method. Surprisingly, the CM method does not perform well for any of the three reference problems.

    4. We conclude that life-history features should be carefully considered when choosing the numerical method for analyzing ecologically realistic size-structured population models.

  • 21.
    Zhang, Lai
    et al.
    Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark.
    Hartvig, Martin
    Knudsen, Kim
    Andersen, Ken Haste
    Size-based predictions of food web patterns2014In: Theoretical Ecology, ISSN 1874-1738, E-ISSN 1874-1746, Vol. 7, no 1, p. 23-33Article in journal (Refereed)
    Abstract [en]

    We employ size-based theoretical arguments to derive simple analytic predictions of ecological patterns and properties of natural communities: size-spectrum expo- nent, maximum trophic level, and susceptibility to invasive species. The predictions are brought about by assuming that an infinite number of species are continuously distributed on a size–trait axis. It is, however, an open question whether such predictions are valid for a food web with a finite num- ber of species embedded in a network structure. We address this question by comparing the size-based predictions to results from dynamic food web simulations with varying species richness. To this end, we develop a new size- and trait-based food web model that can be simplified into an analytically solvable size-based model. We confirm existing solutions for the size distribution and derive novel predic- tions for maximum trophic level and invasion resistance. Our results show that the predicted size-spectrum exponent is borne out in the simulated food webs even with few species, albeit with a systematic bias. The predicted max- imum trophic level turns out to be an upper limit since simulated food webs may have a lower number of trophic levels, especially for low species richness, due to structural constraints. The size-based model possesses an evolutionary stable state and is therefore un-invadable. In contrast, the food web simulations show that all communities, irrespec- tive of number of species, are equally open to invasions. We use these results to discuss the validity of size-based pre- dictions in the light of the structural constraints imposed by food webs. 

  • 22.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Liu, Jia
    Banerjee, Malay
    Hopf and steady state bifurcation analysis in a ratio-dependent predator-prey model2017In: Communications in nonlinear science & numerical simulation, ISSN 1007-5704, E-ISSN 1878-7274, Vol. 44, p. 52-73Article in journal (Refereed)
    Abstract [en]

    In this paper, we perform spatiotemporal bifurcation analysis in a ratio-dependent predator prey model and derive explicit conditions for the existence of non-constant steady states that emerge through steady state bifurcation from related constant steady states. These explicit conditions are numerically verified in details and further compared to those conditions ensuring Turing instability. We find that (1) Turing domain is identical to the parametric domain where there exists only steady state bifurcation, which implies that Turing patterns are stable non-constant steady states, but the opposite is not necessarily true; (2) In non-Turing domain, steady state bifurcation and Hopf bifurcation act in concert to determine the emergent spatial patterns, that is, non-constant steady state emerges through steady state bifurcation but it may be unstable if the destabilising effect of Hopf bifurcation counteracts the stabilising effect of diffusion, leading to non-stationary spatial patterns; (3) Coupling diffusion into an ODE model can significantly enrich population dynamics by inducing alternative non-constant steady states (four different states are observed, two stable and two unstable), in particular when diffusion interacts with different types of bifurcation; (4) Diffusion can promote species coexistence by saving species which otherwise goes to extinction in the absence of diffusion.

  • 23.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Pedersen, Michael
    Lin, Zhigui
    Stability patterns for a size-structured population model and its stage-structured counterpart2015In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 267, p. 109-123Article in journal (Refereed)
    Abstract [en]

    In this paper we compare a general size-structured population model, where a size-structured consumer feeds upon an unstructured resource, to its simplified stage-structured counterpart in terms of equilibrium stability. Stability of the size-structured model is understood in terms of an equivalent delayed system consisting of a renewal equation for the consumer population birth rate and a delayed differential equation for the resource. Results show that the size- and stage-structured models differ considerably with respect to equilibrium stability, although the two models have completely identical equilibrium solutions. First, when adult consumers are superior foragers to juveniles, the size-structured model is more stable than the stagestructured model while the opposite occurs when juveniles are the superior foragers. Second, relatively large juvenile (adult) mortality tends to stabilise (destabilise) the size-structured model but destabilise (stabilise) the stage-structured model. Third, the stability pattern is sensitive to the adult-offspring size ratio in the sizestructured model but much less sensitive in the stage-structured model. Finally, unless the adult-offspring size ratio is sufficiently small, the stage-structured model cannot satisfactorily capture the dynamics of the size-structured model. We conclude that caution must be taken when the stage-structured population model is applied, although it can consistently translate individual life history and stage-specific differences to the population level.

  • 24.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. School of Mathematical Science, Yangzhou University, Si Wang Ting Road, Yangzhou 225002, People’s Republic of China.
    Takahashi, Daisuke
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hartvig, Martin
    Andersen, Ken H.
    Food-web dynamics under climate change2017In: Proceedings of the Royal Society of London. Biological Sciences, ISSN 0962-8452, E-ISSN 1471-2954, Vol. 284, no 1867, article id 20171772Article in journal (Refereed)
    Abstract [en]

    Climate change affects ecological communities through its impact on the physiological performance of individuals. However, the population dynamic of species well inside their thermal niche is also determined by competitors, prey and predators, in addition to being influenced by temperature changes. We use a trait-based food-web model to examine how the interplay between the direct physiological effects from temperature and the indirect effects due to changing interactions between populations shapes the ecological consequences of climate change for populations and for entire communities. Our simulations illustrate how isolated communities deteriorate as populations go extinct when the environment moves outside the species' thermal niches. High-trophic-level species are most vulnerable, while the ecosystem function of lower trophic levels is less impacted. Open communities can compensate for the loss of ecosystem function by invasions of new species. Individual populations show complex responses largely uncorrelated with the direct impact of temperature change on physiology. Such complex responses are particularly evident during extinction and invasion events of other species, where climaticallywell-adapted species may be brought to extinction by the changed food-web topology. Our results highlight that the impact of climate change on specific populations is largely unpredictable, and apparently well-adapted species may be severely impacted.

  • 25.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Thygesen, Uffe Høgsbro
    Banerjee, Malay
    Size-dependent diffusion promotes the emergence of spatiotemporal patterns2014In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 90, no 1, p. 012904-Article in journal (Refereed)
    Abstract [en]

    Spatiotemporal patterns, indicating the spatiotemporal variability of individual abundance, are a pronounced scenario in ecological interactions. Most of the existing models for spatiotemporal patterns treat species as homogeneous groups of individuals with average characteristics by ignoring intraspecific physiological variations at the individual level. Here we explore the impacts of size variation within species resulting from individual ontogeny, on the emergence of spatiotemporal patterns in a fully size-structured population model. We found that size dependency of animal's diffusivity greatly promotes the formation of spatiotemporal patterns, by creating regular spatiotemporal patterns out of temporal chaos. We also found that size-dependent diffusion can substitute large-amplitude base harmonics with spatiotemporal patterns with lower amplitude oscillations but with enriched harmonics. Finally, we found that the single-generation cycle is more likely to drive spatiotemporal patterns compared to predator-prey cycles, meaning that the mechanism of Hopf bifurcation might be more common than hitherto appreciated since the former cycle is more widespread than the latter in case of interacting populations. Due to the ubiquity of individual ontogeny in natural ecosystems we conclude that diffusion variability within populations is a significant driving force for the emergence of spatiotemporal patterns. Our results offer a perspective on self-organized phenomena, and pave a way to understand such phenomena in systems organized as complex ecological networks.

  • 26.
    Zhang, Lai
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Tian, Canrong
    Turing pattern dynamics in an activator-inhibitor system with superdiffusion2014In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, ISSN 1539-3755, E-ISSN 1550-2376, Vol. 90, no 6, p. 062915-Article in journal (Refereed)
    Abstract [en]

    The fractional operator is introduced to an activator-inhibitor system to describe species anomalous superdiffusion. The effects of the superdiffusive exponent on pattern formation and pattern selection are studied. Our linear stability analysis shows that the wave number of the Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons, stripes, and their coexistence are expected. Numerical simulations of the activator-inhibitor model near the stability boundaries confirm our analytical results. Since diffusion rate manifests in both diffusion constant and diffusion exponent, we numerically explore their interactions on the emergence of Turing patterns. When the activator and inhibitor have different superdiffusive exponents, we find that the critical ratio of the diffusion rate of the inhibitor to the activator, required for the formation of the Turing pattern, increases monotonically with the superdiffusive exponent. We conclude that small ratio (than unity) of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the Turing pattern, relative to the normal diffusion.

1 - 26 of 26
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