HOW BIG AND HOW CLOSE? HABITAT PATCH SIZE AND SPACING TO CONSERVE A THREATENED SPECIES

Abstract. We present results of a spatially explicit, individual‐based stochastic dispersal model (HexSim) to evaluate effects of size and spacing of patches of habitat of Northern Spotted Owls (NSO; Strix occidentalis caurina) in Pacific Northwest, USA, to help advise recovery planning efforts. We modeled 31 artificial landscape scenarios representing combinations of NSO habitat cluster size (range 4–49 NSO pairs per cluster) and edge‐to‐edge cluster spacing (range 7–101 km), and an all‐habitat landscape. We ran scenarios using empirical estimates of NSO dispersal dynamics and distances and stage class vital rates (representing current population declines) and under adult survival rates adjusted to achieve an initially stationary population. Results suggested that long‐term (100‐yr) habitat occupancy rates are significantly higher with habitat clusters supporting ≥25 NSO pairs and ≤15 km spacing, and with overall landscapes of ≥35–40% habitat. Although habitat provision is key to NSO recovery, no habitat configuration provided for long‐term population persistence when coupled with currently observed vital rates. Results also suggested a key role of floaters (unpaired, nonterritorial, dispersing owls) in recolonizing vacant habitat, and that the floater population segment becomes increasingly depleted with greater population declines. We suggest additional areas of modeling research on this and other threatened species.     

Spatial Dynamics of a Nonlocal Dispersal Population Model in a Shifting Environment

This paper is concerned with the spatial dynamics of a nonlocal dispersal population model in a shifting environment where the favorable region is shrinking. It is shown that the species becomes extinct in the habitat if the speed of the shifting habitat edge \(c>c^*(\infty )\), while the species persists and spreads along the shifting habitat at an asymptotic speed \(c^*(\infty )\) if \(c<c^*(\infty )\), where \(c^*(\infty )\) is determined by the nonlocal dispersal kernel, diffusion rate and the maximum linearized growth rate. Moreover, we demonstrate that for any given speed of the shifting habitat edge, the model system admits a nondecreasing traveling wave with the wave speed at which the habitat is shifting, which indicates that the extinction wave phenomenon does happen in such a shifting environment.     

On a muted family of bivariate distributions

This paper introduces ‘muted bivariate distribution (MBD)’ as a new family of bivariate distributions possessing the property (P) that the concomitant of nth record value on one variable arising from any distribution of this family is distributed identically as the largest order statistic of a sample of size n arising from the marginal distribution of the same variable. Also if the property P is observed on any distribution, then we illustrate a process of impounding the probability density function (pdf) of the nth record value on one variable resulting due to P with the known form of the marginal pdf on the other variable to build up an appropriate MBD of the parent population. Some properties of the newly defined MBD are further derived. We identify the forms of the pdf of the concomitant of generalized upper record value and the pdf of the concomitant of the smallest order statistic of a sample of size n in order that each of the above pdf’s again can be used to generate MBD of the parent population, alternatively. We further observe MBD as a suitable model for describing an astronomical bivariate dataset relating to the galaxy luminosity measurements made under two conditions on globular clusters. For illustrative purposes, a real-life dataset on acacia trees obtained via concomitant record ranked set sampling methodology is considered as an application of this new family of bivariate distributions.     

Dynamic behavior of a parasite-host model with general incidence

In this paper, we consider the global dynamics of a microparasite model with more general incidences. For the model with the bilinear incidence, Ebert et al. [D. Ebert, M. Lipsitch, K.L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites, American Naturalist 156 (2000) 459-477] observed that parasites can reduce host density, but the extinction of both host population and parasite population occurs only under stochastic perturbations. Hwang and Kuang [T.W. Hwang, Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol. 46 (2003) 17-30] studied the model with the standard incidence and found that the host population may be extinct in the absence of random disturbance. We consider more general incidences that characterize transitions from the bilinear incidence to the standard incidence to simulate behavior changes of populations from random mobility in a fixed area to the mobility with a fixed population density. Using the techniques of Xiao and Ruan [D. Xiao, S. Ruan, Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol. 43 (2001) 268-290], it is shown that parasites can drive the host to extinction only by the standard incidence. The complete classifications of dynamical behaviors of the model are obtained by a qualitative analysis.     

A two-patch prey-predator model with food-gathering activity

In insect ecosystem, the dynamics of prey and predator is regulated by complex interactions between them. Insect pests are spatially aggregated in patches forming a spatial pattern in the environment. An efficient predator dynamically changes its strategies and time for its random search movements to concentrate on higher resource patches based on the benefit of assessment. This food-gathering activity of both prey and predator plays a major role in stabilizing the system by influencing the per unit food consumption. Extending Holling time-budget argument by migration, here we formulate a two patch prey-predator model and show that how several foraging parameters such as handling time, dispersal rate can have important consequences in stability of prey-predator system. Specifically, the ratio between timings that a predator remains mobile in searching and handling their food, is the most important one and simulation on this suggests that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. Thus we conclude that the migration submodel is an important constituent of a spatial predator-prey system. These results are shown to have important implications for possible biological control.     

MODELING A RIGHTS‐BASED APPROACH FOR MANAGING HABITAT IMPACTS OF FISHERIES

ABSTRACT. Limiting adverse consequences of fishing on essential fish habitat has emerged as a key fishery management objective. The conventional approach to providing habitat protection is to create MPAs or marine reserves that prohibit all or certain types of fishing in specific areas. However, there may be more cost‐effective and flexible ways to provide habitat protection. We propose an individual habitat quota (IHQ) system for habitat conservation that would utilize economic incentives to achieve habitat conservation goals cost‐effectively. Individual quotas of habitat impact units (HIU) would be distributed to fishers with an aggregate quota set to maintain a target habitat “stock.” HIU use would be based on a proxy for marginal habitat damage. We use a dynamic, explicitly spatial fishery and habitat simulation model to explore how such a system might work. We examine how outcomes are affected by spatial heterogeneity in the fishery and the scale of habitat regulation. We find that the IHQ system is a highly cost‐effective means of ensuring a given level of habitat protection, but that spatial heterogeneity and the scale of regulation can have significant effects on the distribution of habitat protection.     

Low dimensional homeochaos in coevolving host–parasitoid dimorphic populations: Extinction thresholds under local noise

A discrete time model describing the population dynamics of coevolution between host and parasitoid haploid populations with a dimorphic matching allele coupling is investigated under both determinism and stochastic population disturbances. The role of the properties of the attractors governing the survival of both populations is analyzed considering equal mutation rates and focusing on host and parasitoid growth rates involving chaos. The purely deterministic model reveals a wide range of ordered and chaotic Red Queen dynamics causing cyclic and aperiodic fluctuations of haplotypes within each species. A Ruelle–Takens–Newhouse route to chaos is identified by increasing both host and parasitoid growth rates. From the bifurcation diagram structure and from numerical stability analysis, two different types of chaotic sets are roughly differentiated according to their size in phase space and to their largest Lyapunov exponent: the Confined and Expanded attractors. Under the presence of local population noise, these two types of attractors have a crucial role in the survival of both coevolving populations. The chaotic confined attractors, which have a low largest positive Lyapunov exponent, are shown to involve a very low extinction probability under the influence of local population noise. On the contrary, the expanded chaotic sets (with a higher largest positive Lyapunov exponent) involve higher host and parasitoid extinction probabilities under the presence of noise. The asynchronies between haplotypes in the chaotic regime combined with low dimensional homeochaos tied to the confined attractors is suggested to reinforce the long-term persistence of these coevolving populations under the influence of stochastic disturbances. These ideas are also discussed in the framework of spatially-distributed host–parasitoid populations.     

Evolution of dispersal in a spatially periodic integrodifference model

An integrodifference model describing the reproduction and dispersal of a population is introduced to investigate the evolution of dispersal in a spatially periodic habitat. The dispersal is determined by a kernel function, and the dispersal strategy is defined as the probability of population individuals’ moving to a different habitat. Both conditional and unconditional dispersal strategies are investigated, the distinction being whether dispersal depends on local environmental conditions. For competing unconditional dispersers, we prove that the population with the smaller dispersal probability always prevails. Alternatively, for conditional dispersers, it is shown that the strategy known as ideal free dispersal is both sufficient and necessary for evolutionary stability. These results extend those in the literature for discrete diffusion models in finite patchy landscapes and from reaction–diffusion models.     

Parasite population dynamics within a dynamic host population

Summary We propose a stochastic process model for a parasite population living within a host population. The host population is described by an immigration-death process. The parasite population in one host is an immigration-birth-death-emigration process. The death of all parasites at the moment of death of their host is regarded as emigration. We derive explicit expressions for the distributions of the size of the host population, of the parasite load of one host individual and of the parasite population in the total host population and obtain conditions for the existence of limiting distributions if time is tending to infinity. For particular lifetime distributions of hosts including parasite induced mortality and heterogeneous infection risk we finally derive properties of the limiting distributions.Dedicated to Klaus Krickeberg on the occasion of his 60th birthday     

On global stability of the intra-host dynamics of malaria and the immune system

In this paper we consider an intra-host model for the dynamics of malaria. The model describes the dynamics of the blood stage malaria parasites and their interaction with host cells, in particular red blood cells (RBC) and immune effectors. We establish the equilibrium points of the system and analyze their stability using the theory of competitive systems, compound matrices and stability of periodic orbits. We established that the disease-free equilibrium is globally stable if and only if the basic reproduction number satisfies R₀?1 and the parasite will be cleared out of the host. If R₀>1, a unique endemic equilibrium is globally stable and the parasites persist at the endemic steady state. In the presence of the immune response, the numerical analysis of the model shows that the endemic equilibrium is unstable.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.     

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.     

A bifurcation analysis of stage-structured density dependent integrodifference equations

There is evidence for density dependent dispersal in many stage-structured species, including flour beetles of the genus Tribolium. We develop a bifurcation theory approach to the existence and stability of (non-extinction) equilibria for a general class of structured integrodifference equation models on finite spatial domains with density dependent kernels, allowing for non-dispersing stages as well as partial dispersal. We show that a continuum of such equilibria bifurcates from the extinction equilibrium when it loses stability as the net reproductive number n increases through 1. Furthermore, the stability of the non-extinction equilibria is determined by the direction of the bifurcation. We provide an example to illustrate the theory.     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

脆弱斑块中植物种子的脉冲扩散

考虑了一类脆弱斑块中植物种子的脉冲漂移模型,得到了系统永久持续生存性.在此基础上,利用单调凸算子理论,得到了系统唯一全局渐进稳定的周期解.数值模拟也表明脉冲扩散能挽救脆弱斑块中的植物种群.     

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

ANALYSIS OF SENSITIVITY AND UNCERTAINTY IN AN INDIVIDUAL‐BASED MODEL OF A THREATENED WILDLIFE SPECIES

Sensitivity analysis—determination of how prediction variables affect response variables—of individual‐based models (IBMs) are few but important to the interpretation of model output. We present sensitivity analysis of a spatially explicit IBM (HexSim) of a threatened species, the Northern Spotted Owl (NSO; Strix occidentalis caurina) in Washington, USA. We explored sensitivity to HexSim variables representing habitat quality, movement, dispersal, and model architecture; previous NSO studies have well established sensitivity of model output to vital rate variation. We developed “normative” (expected) model settings from field studies, and then varied the values of ≥ 1 input parameter at a time by ±10% and ±50% of their normative values to determine influence on response variables of population size and trend. We determined time to population equilibration and dynamics of populations above and below carrying capacity. Recovery time from small population size to carrying capacity greatly exceeded decay time from an overpopulated condition, suggesting lag time required to repopulate newly available habitat. Response variables were most sensitive to input parameters of habitat quality which are well‐studied for this species and controllable by management. HexSim thus seems useful for evaluating potential NSO population responses to landscape patterns for which good empirical information is available.     

Spatial Dynamics of a Lattice Lotka-Volterra Competition Model with a Shifting Habitat

In this paper, we concern with the spatial dynamics of the lattice Lotka-Volterra competition system in a shifting habitat. We study the impact of the environmental deterioration rate on the population density under the strong competition condition. Our results show that if the environment deteriorates rapidly, both species will become extinct. However, when the environmental degradation rate is not so fast, the species with slow diffusion will go extinct, while those with fast diffusion will survive. The extinction of species with slow diffusion can be divided into two situations: one is the extinction caused by environmental deterioration faster than its own diffusion speed, the other is the extinction caused by slow diffusion speed under the influence of strong competition.     

Com plete p-elli ptic integrals and a com putation formula of $$\ pi _ p$$π p for $$ p=4$$ p=4

We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.     

An impulsive periodic predator–prey system with Holling type III functional response and diffusion

This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.