A Comparison of the Trojan Y Chromosome Strategy to Harvesting Models for Eradication of Non-Native Species

The Trojan Y Chromosome Strategy (TYC) is a promising eradication method for biological control of non-native species. The strategy works by manipulating the sex ratio of a population through the introduction of \textit{supermales} that guarantee male offspring. In the current manuscript, we compare the TYC method with a pure harvesting strategy. We also analyze a hybrid harvesting model that mirrors the TYC strategy. The dynamic analysis leads to results on stability, global boundedness of solutions and bifurcations of the model. Several conclusions about the different strategies are established via optimal control methods. In particular, the results affirm that either a pure harvesting or hybrid strategy may work better than the TYC method at controlling an invasive species population.Comment: 37 pages, 11 figure

Nonequilibrium plankton community structures in an ecohydrodynamic model system

International audienceDue to the local and global impacts of algae blooms and patchiness on water quality, carbon cycling and climate, models of plankton dynamics are of current interest. In this paper, the temporal and spatial patterns in natural plankton communities are interpreted as transient and stationary nonequilibrium solutions of dynamical nonlinear interaction-diffusion-advection systems. A simple model of phytoplankton-zooplankton dynamics (Scheffer, 1991) is presented in space and time. After summarizing the local properties as multiple stability and oscillations, the emergence of spatial and spatio- temporal patterns is considered, accounting also for diffusion and weak advection. In order to study the emergence and stability of these structures under hydrodynamic forcing, the interaction- diffusion-advection model is coupled to the hydrodynamic equations. It is shown, that the formation of nonequilibrium spatio-temporal density patterns due to the interplay of the deterministic nonlinear biological interactions and physical processes is a rare occurrence in rapidly flowing waters. The two-timing perturbation technique is applied to problems with very rapid single-directed steady flows. A channel under tidal forcing serves as and example for a system with a relatively high detention time of matter. Generally, due to the different time and length scales of planktic interactions, diffusion and transport, initial nonequilibrium plankton patches are simply moved through the system unless the strong hydrodynamic forces do not destroy them before

Evaluating range-expansion models for calculating nonnative species' expansion rate

Species range shifts associated with environmental change or biological invasions are increasingly important study areas. However, quantifying range expansion rates may be heavily influenced by methodology and/or sampling bias. We compared expansion rate estimates of Roesel's bush-cricket (Metrioptera roeselii, Hagenbach 1822), a nonnative species currently expanding its range in south-central Sweden, from range statistic models based on distance measures (mean, median, 95th gamma quantile, marginal mean, maximum, and conditional maximum) and an area-based method (grid occupancy). We used sampling simulations to determine the sensitivity of the different methods to incomplete sampling across the species' range. For periods when we had comprehensive survey data, range expansion estimates clustered into two groups: (1) those calculated from range margin statistics (gamma, marginal mean, maximum, and conditional maximum: similar to 3 km/year), and (2) those calculated from the central tendency (mean and median) and the area-based method of grid occupancy (similar to 1.5 km/year). Range statistic measures differed greatly in their sensitivity to sampling effort; the proportion of sampling required to achieve an estimate within 10% of the true value ranged from 0.17 to 0.9. Grid occupancy and median were most sensitive to sampling effort, and the maximum and gamma quantile the least. If periods with incomplete sampling were included in the range expansion calculations, this generally lowered the estimates (range 16-72%), with exception of the gamma quantile that was slightly higher (6%). Care should be taken when interpreting rate expansion estimates from data sampled from only a fraction of the full distribution. Methods based on the central tendency will give rates approximately half that of methods based on the range margin. The gamma quantile method appears to be the most robust to incomplete sampling bias and should be considered as the method of choice when sampling the entire distribution is not possible

Class of self-limiting growth models in the presence of nonlinear diffusion

The source term in a reaction-diffusion system, in general, does not involve explicit time dependence. A class of self-limiting growth models dealing with animal and tumor growth and bacterial population in a culture, on the other hand are described by kinetics with explicit functions of time. We analyze a reaction-diffusion system to study the propagation of spatial front for these models.Comment: RevTex, 13 pages, 5 figures. To appear in Physical Review

Numerical solutions of random mean square Fisher-KPP models with advection

[EN] This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.Ministerio de Economia y Competitividad, Grant/Award Number: MTM2017-89664-PCasabán Bartual, MC.; Company Rossi, R.; Jódar Sánchez, LA. (2020). Numerical solutions of random mean square Fisher-KPP models with advection. Mathematical Methods in the Applied Sciences. 43(14):8015-8031. https://doi.org/10.1002/mma.5942S801580314314FISHER, R. A. (1937). THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES. Annals of Eugenics, 7(4), 355-369. doi:10.1111/j.1469-1809.1937.tb02153.xBengfort, M., Malchow, H., & Hilker, F. M. (2016). The Fokker–Planck law of diffusion and pattern formation in heterogeneous environments. Journal of Mathematical Biology, 73(3), 683-704. doi:10.1007/s00285-016-0966-8Okubo, A., & Levin, S. A. (2001). Diffusion and Ecological Problems: Modern Perspectives. Interdisciplinary Applied Mathematics. doi:10.1007/978-1-4757-4978-6SKELLAM, J. G. (1951). RANDOM DISPERSAL IN THEORETICAL POPULATIONS. Biometrika, 38(1-2), 196-218. doi:10.1093/biomet/38.1-2.196Aronson, D. G., & Weinberger, H. F. (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Partial Differential Equations and Related Topics, 5-49. doi:10.1007/bfb0070595Aronson, D. ., & Weinberger, H. . (1978). Multidimensional nonlinear diffusion arising in population genetics. Advances in Mathematics, 30(1), 33-76. doi:10.1016/0001-8708(78)90130-5Weinberger, H. F. (2002). On spreading speeds and traveling waves for growth and migration models in a periodic habitat. Journal of Mathematical Biology, 45(6), 511-548. doi:10.1007/s00285-002-0169-3Weinberger, H. F., Lewis, M. A., & Li, B. (2007). Anomalous spreading speeds of cooperative recursion systems. Journal of Mathematical Biology, 55(2), 207-222. doi:10.1007/s00285-007-0078-6Liang, X., & Zhao, X.-Q. (2006). Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Communications on Pure and Applied Mathematics, 60(1), 1-40. doi:10.1002/cpa.20154E. Fitzgibbon, W., Parrott, M. E., & Webb, G. (1995). Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete & Continuous Dynamical Systems - A, 1(1), 35-57. doi:10.3934/dcds.1995.1.35Kinezaki, N., Kawasaki, K., & Shigesada, N. (2006). Spatial dynamics of invasion in sinusoidally varying environments. Population Ecology, 48(4), 263-270. doi:10.1007/s10144-006-0263-2Jin, Y., Hilker, F. M., Steffler, P. M., & Lewis, M. A. (2014). Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows. Bulletin of Mathematical Biology, 76(7), 1522-1565. doi:10.1007/s11538-014-9957-3Faou, E. (2009). Analysis of splitting methods for reaction-diffusion problems using stochastic calculus. Mathematics of Computation, 78(267), 1467-1483. doi:10.1090/s0025-5718-08-02185-6Doering, C. R., Mueller, C., & Smereka, P. (2003). Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A: Statistical Mechanics and its Applications, 325(1-2), 243-259. doi:10.1016/s0378-4371(03)00203-6Siekmann, I., Bengfort, M., & Malchow, H. (2017). Coexistence of competitors mediated by nonlinear noise. The European Physical Journal Special Topics, 226(9), 2157-2170. doi:10.1140/epjst/e2017-70038-6McKean, H. P. (1975). Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov. Communications on Pure and Applied Mathematics, 28(3), 323-331. doi:10.1002/cpa.3160280302Berestycki, H., & Nadin, G. (2012). Spreading speeds for one-dimensional monostable reaction-diffusion equations. Journal of Mathematical Physics, 53(11), 115619. doi:10.1063/1.4764932Cortés, J. C., Jódar, L., Villafuerte, L., & Villanueva, R. J. (2007). Computing mean square approximations of random diffusion models with source term. Mathematics and Computers in Simulation, 76(1-3), 44-48. doi:10.1016/j.matcom.2007.01.020Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Casabán, M.-C., Cortés, J.-C., & Jódar, L. (2016). Solving linear and quadratic random matrix differential equations: A mean square approach. Applied Mathematical Modelling, 40(21-22), 9362-9377. doi:10.1016/j.apm.2016.06.017Sarmin, E. N., & Chudov, L. A. (1963). On the stability of the numerical integration of systems of ordinary differential equations arising in the use of the straight line method. USSR Computational Mathematics and Mathematical Physics, 3(6), 1537-1543. doi:10.1016/0041-5553(63)90256-8Sanz-Serna, J. M., & Verwer, J. G. (1989). Convergence analysis of one-step schemes in the method of lines. Applied Mathematics and Computation, 31, 183-196. doi:10.1016/0096-3003(89)90118-5Calvo, M. P., de Frutos, J., & Novo, J. (2001). Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations. Applied Numerical Mathematics, 37(4), 535-549. doi:10.1016/s0168-9274(00)00061-1Cox, S. M., & Matthews, P. C. (2002). Exponential Time Differencing for Stiff Systems. Journal of Computational Physics, 176(2), 430-455. doi:10.1006/jcph.2002.6995De la Hoz, F., & Vadillo, F. (2016). Numerical simulations of time-dependent partial differential equations. Journal of Computational and Applied Mathematics, 295, 175-184. doi:10.1016/j.cam.2014.10.006Company, R., Egorova, V. N., & Jódar, L. (2018). Conditional full stability of positivity-preserving finite difference scheme for diffusion–advection-reaction models. Journal of Computational and Applied Mathematics, 341, 157-168. doi:10.1016/j.cam.2018.02.031Kaczorek, T. (2002). Positive 1D and 2D Systems. Communications and Control Engineering. doi:10.1007/978-1-4471-0221-2Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences. doi:10.1007/978-1-4612-5561-

Wannier functions analysis of the nonlinear Schr\"{o}dinger equation with a periodic potential

In the present Letter we use the Wannier function basis to construct lattice approximations of the nonlinear Schr\"{o}dinger equation with a periodic potential. We show that the nonlinear Schr\"{o}dinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to Bose-Einstein condensate theory as well as to other physical systems like, for example, electromagnetic wave propagation in nonlinear photonic crystals.Comment: 5 pages, 1 figure, submitted to Phys. Rev.

Traveling wave solutions for a predator-prey system with Sigmoidal response function

We study the existence of traveling wave solutions for a diffusive predator-prey system. The system considered in this paper is governed by a Sigmoidal response function which is more general than those studied previously. Our method is an improvement to the original method introduced in the work of Dunbar \cite{Dunbar1,Dunbar2}. A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in \cite{Dunbar1,Dunbar2}. The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski's theorem which is much simpler than the extended version in Dunbar's work