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In this paper, we derive simple analytical bounds for solutions of, and use them for estimating trajectories following Lotka-Volterra-type integrals. We show how our results give estimates for the Lambert W function as well as for trajectories of general predator-prey systems, including, for example, Rosenzweig-MacArthur equations.

We prove estimates for the maximal and minimal predator and prey populations on the unique limit cycle in a standard predatorprey system. Our estimates are valid when the cycle exhibits small predator and prey abundances and large amplitudes. The proofs consist of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates, and should be of independent interest. This study generalizes results proved by the authors in [11] to a wider class of systems and, in addition, it gives simpler proofs of some already known estimates.

We use a dynamic programming approach to construct management strategies for a hydropower plant with a dam and a continuously adjustable unit. Along the way, we estimate unknown variables via simple models using historical data and forecasts. Our suggested scheme achieves on average 97.5% of the theoretical maximum (optimal strategy when knowing the future) with small computational complexity. We also apply our scheme to a run-of-river hydropower plant and compare the strategies and results to the much more involved PDE-based optimal switching method studied in [12]; this comparison shows that our simple approach may be preferable when the forecast is good enough.

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x, u, Du, D2u) = 0 under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

Suppose that p∈(1,∞], ν∈[1/2,∞), Sν= {(x1, x2)∈R2\{(0, 0)}:|φ|< π/2ν}, where φ is the polar angle of (x1, x2). Let R>0 and ωp(x) be the p-harmonic measure of ∂B(0,R)∩Sν at x with respect to B(0,R)∩Sν. We prove that there exists a constant C such that whenever x∈B(0,R)∩S2ν and where the exponent k(ν, p) is given explicitly as a function of ν and p. Using this estimate we derive local growth estimates for p-sub- and p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmén-Lindelöf theorem for p-subharmonic functions in the unbounded sector Sν. Moreover, if p=∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in Rn. Finally, when ν∈(1/2,∞) and p∈(1,∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on ∂Sν.

We consider a Rosenzweig–MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are of independent interest.

We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the form F(x,u,Du,D²u)=0 in terms of solutions to ordinary differential equations built upon assumptions on F. Using this characterization we derive several sharp Phragmen–Lindelöf-type theorems for certain classes of well known PDEs. The equation need not be uniformly elliptic nor homogeneous and we obtain results both in case the subsolution is bounded or unbounded. Among our results we retrieve classical estimates in the halfspace for p-subharmonic functions and extend those to more general equations; we prove sharp growth estimates, in terms of k and the asymptotic behavior of ∫₀^RC(s)ds, for subsolutions of equations allowing for sublinear growth in the gradient of the form C(|x|)|Du|^k with k≥1; we establish a Phragmen–Lindelöf theorem for weak subsolutions of the variable exponent p-Laplace equation in halfspaces, 1<p(x)<∞, p(x)∈C¹, of which we conclude sharpness by finding the "slowest growing" p(x)-harmonic function together with its corresponding family of p(x)-exponents. The paper ends with a discussion of our results from the point of view of a spatially dependent diffusion problem.

The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a conceptual production planning problem related to hydropower. In particular, we study two examples of small run-of-river hydropower plants and provide an outline of how optimal switching can be used to create fully automatic production schemes for these. Along the way, we create a new model for random flow of water based on stochastic differential equations and fit this model to historical data. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 and 5 %, respectively, in a long term investigation of the two plants considered.