Dynamic Sheltering Problem and An Existence Theorem for Admissible Strategies

This paper is devoted to a new class of dynamic problems, originally motivated by the protection issues when a spreading disaster occurs. In the propagation model, the invaded region is described by the reachable set of differential inclusions. To protect an assigned habitat against the disaster, an artificial barrier is constructed to shield the habitat in real time, where the barrier is characterized as one-dimensional rectifiable set in mathematical terms and a dynamic strategy is determined by the barrier constructed over time. In the paper, we start by showing the motivation of dynamic sheltering problems and then describe the mathematical models. Afterwards, we restrict the analysis to an isotropic case, and derive the minimum speed that ensures the existence of an admissible sheltering strategy. As the main conclusion, it is proved that, given a convex habitat, among all barriers each consisting of a Jordan curve, there exists an admissible sheltering strategy, if and only if the construction speed is greater than that determined by the boundary curve of the habitat. After stating the main theorem, an illustrative example is carefully examined, which provides insights into the existence theorem and clarifies several common misconceptions.     

Does movement toward better environments always benefit a population?

We study the effects of advection along environmental gradients on logistic reaction-diffusion models for population growth. The local population growth rate is assumed to be spatially inhomogeneous, and the advection is taken to be a multiple of the gradient of the local population growth rate. It is also assumed that the boundary acts as a reflecting barrier to the population. We show that the effects of such advection depend crucially on the shape of the habitat of the population: if the habitat is convex, the movement in the direction of the gradient of the growth rate is always beneficial to the population, while such advection could be harmful for certain non-convex habitats.     

Baire category and boundary value problems for ordinary and partial differential inclusions under Carathéodory assumptions

By using the Baire category method we prove an existence result for boundary value problem of Dirichlet type, for non-convex ordinary differential inclusions under Caratheodory assumptions. By counterexamples we show that an analogous existence result is non-valid for non-convex partial differential inclusions.     

Planar Location Problems with Block Distance and Barriers

This paper considers one facility planar location problems using block distance and assuming barriers to travel. Barriers are defined as generalized convex sets relative to the block distance. The objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a non-convex objective function. The problem is solved by modifying the barriers to obtain an equivalent problem and by decomposing the feasible region into a polynomial number of convex subsets on which the objective function is convex. It is shown that solving a planar location problem with block distance and barriers requires at most a polynomial amount of additional time over solving the same problem without barriers.     

Determining optimal treatment rate after a disaster

From the standpoint of medical services, a disaster is a calamitous event resulting in an unexpected number of casualties that exceeds the therapeutic capacities of medical services. In these situations, effective medical response plays a crucial role in saving life. To model medical rescue activities, a two-priority non-preemptive S-server, and a finite capacity queueing system is considered. After constructing Chapman–Kolmogorov differential equations, Pontryagin's minimum principle is used to calculate optimal treatment rates for each priority class. The performance criterion is to minimize both the expected value of the square of the difference between the number of servers and the number of patients in the system, and also the cost of serving these patients over a determined time period. The performance criterion also includes a final time cost related to deviations from the determined value of the desired queue length. The two point boundary value problem is numerically solved for different arrival rate patterns and selected parameters.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

Variational principles with integrands defined through a minimum

We revisit a systematic approach for the computation and analysis of the convex hull of non-convex integrands defined through the minimum of convex densities. It consists in reformulating the non-convex variational problem as a double minimization and exploiting appropriately the nature of optimality of the inner minimization. This requires gradient Young measures in the vector case, even if the initial problem was scalar, as the full problem is recast through the computation of a certain quasiconvexification. We illustrate this strategy by looking at two typical non-convex scalar problems. We hope to address vector problems in the near future.     

Cavity Volume and Free Energy in Many-Body Systems

Most models for the spread of an invasive species into a new environment are based on Fisher’s reaction–diffusion equation. They assume that habitat quality is independent of the presence or absence of the invading population. Ecosystem engineers are species that modify their environment to make it (more) suitable for them. A potentially more appropriate modeling approach for such an invasive species is to adapt the well-known Stefan problem of melting ice. Ahead of the front, the habitat is unsuitable for the species (the ice); behind the front, the habitat is suitable (the open water). The engineering action of the population moves the boundary ahead (the melting). This approach leads to a free boundary problem. In this paper, we study the well-posedness of a novel free boundary model for the spread of ecosystem engineers that was recently derived from an individual random walk model. The Stefan condition for the moving boundary is replaced by a biologically derived two-sided condition that models the movement behavior of individuals at the boundary as well as the process by which the population moves the boundary to expand their territory. Our proofs consist of several new and novel ideas that can be used in broader contexts. We assign a convex functional to this problem so that the evolution system governed by this convex potential is exactly the system of evolution equations describing the above model. We then apply variational and fixed-point methods to deal with this free boundary problem.

     

SUSTAINABLE FOREST MANAGEMENT FOR OPTIMIZING MULTISPECIES WILDLIFE HABITAT: A COASTAL DOUGLAS-FIR EXAMPLE

Wildlife species viability optimization models are developed to convert a given set of initial forest conditions, through a combination of natural growth and management treatments, to a forest system which addresses the joint habitat needs of multispecies populations over time. A linear model of forest cover and wildlife populations is used to form a system of forest management control variables for wildlife habitat modification. The paper examines two objective functions coupled to this system for optimizing sustainable joint species viability. The first maximizes the product of periodic joint viabilities over all time periods, focusing management resources on long-term equilibria, with less emphasis on conversion strategy. The second iteratively maximizes the minimum periodic joint viability over all time periods. This focuses management resources on the most limiting time periods, typically the conversion phase periods. Both objective functions resulted in either point or cyclic equilibria, with cycle lengths equal to minimum forest treatment ages. A third objective, based on maximizing the minimum individual species periodic viability is used to examine single species emphasis. Examples are developed through a case study of 92 vertebrate species found in coastal Douglas-fir stands of northwestern California.     

Boundary Value Problems for Riemann–Liouville Fractional Differential Inclusions with Nonlocal Hadamard Fractional Integral Conditions

In this paper, we study a new class of boundary value problems from a fractional differential inclusion of Riemann–Liouville type and nonlocal Hadamard fractional integral boundary conditions. Some new existence results for convex as well as non-convex multi-valued maps are obtained using standard fixed point theorems. The obtained results are illustrated by examples.     

On the construction and investigation of solutions of boundary value problems of thermoviscoelasticity

The uncoupled mixed boundary value problem of thermoviscoelasticity is considered in a quasistatic formulation. The temperature distribution is assumed nonstationary and inhomogeneous. The influence of the temperature on the viscoelastic properties of the material is taken into account by the introduction of a reduced time. The equations of state of the material are written in differential form as a system of kinetic equations in some tensor-type strain parameters. The system mentioned is equivalent to a Volterra integral equation with kernel in the form of a sum of exponents. The differential approach used is apparently more convenient for numerical realization /1/ (especially in nonuniform problems) and results in a substantially different mathematical formulation as compared with that based on the integral form of writing the equations of state investigated in /2,3/. Precisely for going over to the boundary value problem are the kinetic differential equations converted into an operator differential equation in Hubert space. The existence, uniqueness, and stability of the solution of the problem formulated are established, and conditions for the convergence of the Galerkin approximations and the stability of the difference approximations in time are formulated.     

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

Computing minimum-area rectilinear convex hull and L-shape

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O(n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O(n²) time and O(n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.     

Location with acceleration–deceleration distance

In this paper we investigate a model where travel time is not necessarily proportional to the distance. Every trip starts at speed zero, then the vehicle accelerates to a cruising speed, stays at the cruising speed for a portion of the trip and then decelerates back to a speed of zero. We define a time equivalent distance which is equal to the travel time multiplied by the cruising speed. This time equivalent distance is referred to as the acceleration–deceleration (A–D) distance. We prove that every demand point is a local minimum for the Weber problem defined by travel time rather than distance. We propose a heuristic approach employing the generalized Weiszfeld algorithm and an optimal approach applying the Big Triangle Small Triangle global optimization method. These two approaches are very efficient and problems of 10,000 demand points are solved in about 0.015 seconds by the generalized Weiszfeld algorithm and in about 1 minute by the BTST technique. When the generalized Weiszfeld algorithm was repeated 1000 times, the optimal solution was found at least once for all test problems.     

Global optimization for the sum of generalized polynomial fractional functions

In this paper, a branch and bound approach is proposed for global optimization problem (P) of the sum of generalized polynomial fractional functions under generalized polynomial constraints, which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. By utilizing an equivalent problem and some linear underestimating approximations, a linear relaxation programming problem of the equivalent form is obtained. Consequently, the initial non-convex nonlinear problem (P) is reduced to a sequence of linear programming problems through successively refining the feasible region of linear relaxation problem. The proposed algorithm is convergent to the global minimum of the primal problem by means of the solutions to a series of linear programming problems. Numerical results show that the proposed algorithm is feasible and can successfully be used to solve the present problem (P).     

Minimum principle for systems with boundary conditions and cost having Stieltjes-type integrals

Pontryagin's minimum principle for nonlinear differential systems with Stieltjes boundary constraints and performance index with an integral of the Lebesgue-Stieltjes type is derived. The case of infinite time is also considered.     

A diffusive Lotka–Volterra model with Robin boundary condition and sign-changing growth rates in time-periodic environment

In this paper, a Lotka–Volterra model with Robin and free boundary conditions is considered in the heterogeneous time-periodic environment. We mainly consider the changes of local growth rates of native and invasive species that might be negative in some large regions. We study the spreading–vanishing dichotomy. When vanishing occurs, a native species cannot spread successfully as time goes to infinity. However, for an invasive species, in the long run, either it will go extinct or converge to the unique positive solution of time-periodic boundary value problem of logistic equation. When spreading occurs, both native and invasive species have upper and lower bounds. We also obtain the criteria for spreading and vanishing, and estimate of the asymptotic spreading speed.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.