A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation

This paper aims to develop a meshless approach based on the Point Interpolation Method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach.

Accounting for management costs in sensitivity analyses of matrix population models

Traditional sensitivity and elasticity analyses of matrix population models have been used to inform management decisions, but they ignore the economic costs of manipulating vital rates. For example, the growth rate of a population is often most sensitive to changes in adult survival rate, but this does not mean that increasing that rate is the best option for managing the population because it may be much more expensive than other options. To explore how managers should optimize their manipulation of vital rates, we incorporated the cost of changing those rates into matrix population models. We derived analytic expressions for locations in parameter space where managers should shift between management of fecundity and survival, for the balance between fecundity and survival management at those boundaries, and for the allocation of management resources to sustain that optimal balance. For simple matrices, the optimal budget allocation can often be expressed as simple functions of vital rates and the relative costs of changing them. We applied our method to management of the Helmeted Honeyeater (Lichenostomus melanops cassidix; an endangered Australian bird) and the koala (Phascolarctos cinereus) as examples. Our method showed that cost-efficient management of the Helmeted Honeyeater should focus on increasing fecundity via nest protection, whereas optimal koala management should focus on manipulating both fecundity and survival simultaneously. These findings are contrary to the cost-negligent recommendations of elasticity analysis, which would suggest focusing on managing survival in both cases. A further investigation of Helmeted Honeyeater management options, based on an individual-based model incorporating density dependence, spatial structure, and environmental stochasticity, confirmed that fecundity management was the most cost-effective strategy. Our results demonstrate that decisions that ignore economic factors will reduce management efficiency. ©2006 Society for Conservation Biology.

Making robust decisions for conservation with restricted money and knowledge

1. In conservation decision-making, we operate within the confines of limited funding. Furthermore, we often assume particular relationships between management impact and our investment in management. The structure of these relationships, however, is rarely known with certainty - there is model uncertainty. We investigate how these two fundamentally limiting factors in conservation management, money and knowledge, impact optimal decision-making. 2. We use information-gap decision theory to find strategies for maximizing the number of extant subpopulations of a threatened species that are most immune to failure due to model uncertainty. We thus find a robust framework for exploring optimal decision-making. 3. The performance of every strategy decreases as model uncertainty increases. 4. The strategy most robust to model uncertainty depends not only on what performance is perceived to be acceptable but also on available funding and the time horizon over which extinction is considered. 5. Synthesis and applications. We investigate the impact of model uncertainty on robust decision-making in conservation and how this is affected by available conservation funding. We show that subpopulation triage can be a natural consequence of robust decision-making. We highlight the need for managers to consider triage not as merely giving up, but as a tool for ensuring species persistence in light of the urgency of most conservation requirements, uncertainty and the poor state of conservation funding. We illustrate this theory by a specific application to allocation of funding to reduce poaching impact on the Sumatran tiger Panthera tigris sumatrae in Kerinci Seblat National Park. © 2008 The Authors.

Optimal allocation of conservation effort among subpopulations of a threatened species: How important is patch quality?

Money is often a limiting factor in conservation, and attempting to conserve endangered species can be costly. Consequently, a framework for optimizing fiscally constrained conservation decisions for a single species is needed. In this paper we find the optimal budget allocation among isolated subpopulations of a threatened species to minimize local extinction probability. We solve the problem using stochastic dynamic programming, derive a useful and simple alternative guideline for allocating funds, and test its performance using forward simulation. The model considers subpopulations that persist in habitat patches of differing quality, which in our model is reflected in different relationships between money invested and extinction risk. We discover that, in most cases, subpopulations that are less efficient to manage should receive more money than those that are more efficient to manage, due to higher investment needed to reduce extinction risk. Our simple investment guideline performs almost as well as the exact optimal strategy. We illustrate our approach with a case study of the management of the Sumatran tiger, Panthera tigris sumatrae, in Kerinci Seblat National Park (KSNP), Indonesia. We find that different budgets should be allocated to the separate tiger subpopulations in KSNP. The subpopulation that is not at risk of extinction does not require any management investment. Based on the combination of risks of extinction and habitat quality, the optimal allocation for these particular tiger subpopulations is an unusual case: subpopulations that occur in higher-quality habitat (more efficient to manage) should receive more funds than the remaining subpopulation that is in lower-quality habitat. Because the yearly budget allocated to the KSNP for tiger conservation is small, to guarantee the persistence of all the subpopulations that are currently under threat we need to prioritize those that are easier to save. When allocating resources among subpopulations of a threatened species, the combined effects of differences in habitat quality, cost of action, and current subpopulation probability of extinction need to be integrated. We provide a useful guideline for allocating resources among isolated subpopulations of any threatened species. © 2010 by the Ecological Society of America.

Robust conservation decision-making

Decision-making for conservation is conducted within the margins of limited funding. Furthermore, to allocate these scarce resources we make assumptions about the relationship between management impact and expenditure. The structure of these relationships, however, is rarely known with certainty. We present a summary of work investigating the impact of model uncertainty on robust decision-making in conservation and how this is affected by available conservation funding. We show that achieving robustness in conservation decisions can require a triage approach, and emphasize the need for managers to consider triage not as surrendering but as rational decision making to ensure species persistence in light of the urgency of the conservation problems, uncertainty, and the poor state of conservation funding. We illustrate this theory by a specific application to allocation of funding to reduce poaching impact on the Sumatran tiger Panthera tigris sumatrae in Kerinci Seblat National Park, Indonesia. To conserve our environment, conservation managers must make decisions in the face of substantial uncertainty. Further, they must deal with the fact that limitations in budgets and temporal constraints have led to a lack of knowledge on the systems we are trying to preserve and on the benefits of the actions we have available (Balmford & Cowling 2006). Given this paucity of decision-informing data there is a considerable need to assess the impact of uncertainty on the benefit of management options (Regan et al. 2005). Although models of management impact can improve decision making (e.g.Tenhumberg et al. 2004), they typically rely on assumptions around which there is substantial uncertainty. Ignoring this 'model uncertainty', can lead to inferior decision-making (Regan et al. 2005), and potentially, the loss of the species we are trying to protect. Current methods used in ecology allow model uncertainty to be incorporated into the model selection process (Burnham & Anderson 2002; Link & Barker 2006), but do not enable decision-makers to assess how this uncertainty would change a decision. This is the basis of information-gap decision theory (info-gap); finding strategies most robust to model uncertainty (Ben-Haim 2006). Info-gap has permitted conservation biology to make the leap from recognizing uncertainty to explicitly incorporating severe uncertainty into decision-making. In this paper we present a summary of McDonald-Madden et al (2008a) who use an info-gap framework to address the impact of uncertainty in the functional representations of biological systems on conservation decision-making. Furthermore, we highlight the importance of two key elements limiting conservation decision-making - funding and knowledge - and how they interact to influence the best management strategy for a threatened species. Copyright © ASCE 2011.

Should we implement monitoring or research for conservation?

[Letter to editor, brief commentary or brief communication ]

Subpopulation triage: How to allocate conservation effort among populations

Threatened species often exist in a small number of isolated subpopulations. Given limitations on conservation spending, managers must choose from strategies that range from managing just one subpopulation and risking all other subpopulations to managing all subpopulations equally and poorly, thereby risking the loss of all subpopulations. We took an economic approach to this problem in an effort to discover a simple rule of thumb for optimally allocating conservation effort among subpopulations. This rule was derived by maximizing the expected number of extant subpopulations remaining given n subpopulations are actually managed. We also derived a spatiotemporally optimized strategy through stochastic dynamic programming. The rule of thumb suggested that more subpopulations should be managed if the budget increases or if the cost of reducing local extinction probabilities decreases. The rule performed well against the exact optimal strategy that was the result of the stochastic dynamic program and much better than other simple strategies (e.g., always manage one extant subpopulation or half of the remaining subpopulation). We applied our approach to the allocation of funds in 2 contrasting case studies: reduction of poaching of Sumatran tigers (Panthera tigris sumatrae) and habitat acquisition for San Joaquin kit foxes (Vulpes macrotis mutica). For our estimated annual budget for Sumatran tiger management, the mean time to extinction was about 32 years. For our estimated annual management budget for kit foxes in the San Joaquin Valley, the mean time to extinction was approximately 24 years. Our framework allows managers to deal with the important question of how to allocate scarce conservation resources among subpopulations of any threatened species. © 2008 Society for Conservation Biology.

Using conservation evidence to guide management

Almost 10 years ago, Pullin and Knight (2001) called for an “effectiveness revolution in conservation” to be enabled by the systematic evaluation of evidence for conservation decision making. Drawing from the model used in clinicalmedicine, they outlined the concept of “evidencebased conservation” in which existing information, or evidence, from relevant and rigorous research is compiled and analyzed in a systematic manner to inform conservation actions (Cochrane 1972). The promise of evidencebased conservation has generated significant interest; 25 systematic reviews have been completed since 2004 and dozens are underway (Collaboration for Environmental Evidence 2010). However we argue that an “effectiveness revolution” (Pullin & Knight 2001) in conservation will not be possible unless mechanisms are devised for incorporating the growing evidence base into decision frameworks. For conservation professionals to accomplish the missions of their organizations they must demonstrate that their actions actually achieve objectives (Pullin & Knight 2009). Systematic evaluation provides a framework for objectively evaluating the effectiveness of actions. To leverage the benefit of these evaluations, we need resource-allocation systems that are responsive to their outcomes. The allocation of conservation resources is often the product of institutional priorities or reliance on intuition (Sutherland et al. 2004; Pullin & Knight 2005; Cook et al. 2010). We highlight the NICE technologyappraisal process because it provides an example of formal integration of systematic-evidence evaluation with provision of guidance for action. The transparent process, which clearly delineates costs and benefits of each alternative action, could also provide the public with new insight into the environmental effects of different decisions. This insight could stimulate a wider discussion about investment in conservation by demonstrating how changes in funding might affect the probability of achieving conservation objectives. ©2010 Society for Conservation Biology

Ecosystem management along ephemeral rivers: Trading off socio-economic water supply and vegetation conservation under flood regime uncertainty

In ecosystems driven by water availability, plant community dynamics depend on complex interactions between vegetation, hydrology, and human water resources use. Along ephemeral rivers—where water availability is erratic—vegetation and people are particularly vulnerable to changes in each other's water use. Sensible management requires that water supply be maintained for people, while preserving ecosystem health. Meeting such requirements is challenging because of the unpredictable water availability. We applied information gap decision theory to an ecohydrological system model of the Kuiseb River environment in Namibia. Our aim was to identify the robustness of ecosystem and water management strategies to uncertainties in future flood regimes along ephemeral rivers. We evaluated the trade-offs between alternative performance criteria and their robustness to uncertainty to account for both (i) human demands for water supply and (ii) reducing the risk of species extinction caused by water mining. Increasing uncertainty of flood regime parameters reduced the performance under both objectives. Remarkably, the ecological objective (species coexistence) was more sensitive to uncertainty than the water supply objective. However, within each objective, the relative performance of different management strategies was insensitive to uncertainty. The ‘best’ management strategy was one that is tuned to the competitive species interactions in the Kuiseb environment. It regulates the biomass of the strongest competitor and, thus, at the same time decreases transpiration, thereby increasing groundwater storage and reducing pressure on less dominant species. This robust mutually acceptable strategy enables species persistence without markedly reducing the water supply for humans. This study emphasises the utility of ecohydrological models for resource management of water-controlled ecosystems. Although trade-offs were identified between alternative performance criteria and their robustness to uncertain future flood regimes, management strategies were identified that help to secure an ecologically sustainable water supply.

Analysis of passenger group behaviour and its impact on passenger flow using an agent-based model

Group interaction within crowds is a common phenomenon and has great influence on pedestrian behaviour. This paper investigates the impact of passenger group dynamics using an agent-based simulation method for the outbound passenger process at airports. Unlike most passenger-flow models that treat passengers as individual agents, the proposed model additionally incorporates their group dynamics as well. The simulation compares passenger behaviour at airport processes and discretionary services under different group formations. Results from experiments (both qualitative and quantitative) show that incorporating group attributes, in particular, the interactions with fellow travellers and wavers can have significant influence on passengers activity preference as well as the performance and utilisation of services in airport terminals. The model also provides a convenient way to investigate the effectiveness of airport space design and service allocations, which can contribute to positive passenger experiences. The model was created using AnyLogic software and its parameters were initialised using recent research data published in the literature.

High order unconditionally stable difference schemes for the Riesz space-fractional telegraph equation

In this paper, a class of unconditionally stable difference schemes based on the Pad´e approximation is presented for the Riesz space-fractional telegraph equation. Firstly, we introduce a new variable to transform the original dfferential equation to an equivalent differential equation system. Then, we apply a second order fractional central difference scheme to discretise the Riesz space-fractional operator. Finally, we use (1, 1), (2, 2) and (3, 3) Pad´e approximations to give a fully discrete difference scheme for the resulting linear system of ordinary differential equations. Matrix analysis is used to show the unconditional stability of the proposed algorithms. Two examples with known exact solutions are chosen to assess the proposed difference schemes. Numerical results demonstrate that these schemes provide accurate and efficient methods for solving a space-fractional hyperbolic equation.

A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains

Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.