Rare and elusive species are seldom the first choice of model for the study of ecological questions, yet rarity and elusiveness often emerge from ecological processes. One of these processes is interspecific killing, the most extreme form of interference competition among carnivores. Subdominant species can avoid falling victim to other carnivores through spatial and/or temporal separation. The smallest carnivore species, including members of the Mustelidae, are typically the most threatened by other predators but are also exceedingly challenging to study in the wild. As a consequence, we have only limited knowledge of how the most at-risk members of carnivore communities deal with being both hunters and hunted. We explored whether activity and space use of a little-known small carnivore, the Altai mountain weasel Mustela altaica, reflect the activity and distribution of its main prey, pika Ochotona sp., and two sympatric predators, the stone marten Martes foina and the red fox Vulpes vulpes. Spatial and temporal patterns of photographic captures in Pakistan's northern mountains suggest that weasels may cope with being both predator and prey by frequenting areas used by pikas while exhibiting diurnal activity that contrasts with that of the mostly nocturnal/crepuscular stone marten and red fox. Camera trap studies are now common and are staged in many different ecosystems. The data yielded offer an opportunity not only to fill knowledge gaps concerning less-studied species but also to non-invasively test ecological hypotheses linked with rarity and elusiveness.
It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations.
A localized radial basis function meshless method is applied to approximate a nonlinear biological population model with highly satisfactory results. The method approximates the derivatives at every point corresponding to their local support domain. The method is well suited for arbitrary domains. Compared to the finite element and element free Galerkin methods, no integration tool is required. Four examples are demonstrated to check the efficiency and accuracy of the method. The results are compared with an exact solution and other methods available in literature.
This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically
In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.
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