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A novel general stability analysis scheme based on a non-Lyapunov framework is explored. Several easy-to-check sufficient conditions for exponential p-stability are formulated in terms of M-matrices. Stability analysis of applied second-order Ito equations with delay is provided as well. The linearization technique, in combination with the tests obtained in this paper, can be used for local stability analysis of a wide class of nonlinear stochastic differential equations.

A newly presented McKendrick–Von Foerster model with a stochastically perturbed mortality rate is examined. A transformation method converting the model with nonlocal boundary conditions into a system of stochastic functional differential equations is offered. The method could be viewed as analogous to the one which is widely used for such type of deterministic problems. The derived stochastic functional differential equations yield multiple classic population models with ‘naturally born’ stochasticity, including delayed Nicholson’s blowflies, general recruitment and models with cannibalism, which by itself could be objects of future analysis and applications. The work of the first and the third author was partially supported by the grant #2390701 of the Research Council of Norway. This is an electronic version of an article originally published as: Ponosov, A., Idels, L., & Kadiev, R. (2020). Stochastic McKendrick-Von Foerster models with applications. Physica A: Statistical Mechanics and its Applications, 537(1), 1-14. DOI: 10.1016/j.physa.2019.122641 Physica A: Statistical Mechanics and its Applications is published by Elsevier and can be found online at: https://www.sciencedirect.com/journal/physica-a-statistical-mechanics-and-its-applications. The formal publication of this article can be found at: https://dx.doi.org/10.1016/j.physa.2019.122641. This is an open access article, and is available under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). Article 122641 https://viurrspace.ca/bitstream/handle/10613/22969/Idels.PhysicaA.pdf?sequence=3

For the nonlinear second order Lienard-type equations with time-varying delays $$ \ddot{x}(t)+\sum_{k=1}^m f_k(t,x(t),\dot{x}(g_k(t)))+\sum_{k=1}^l s_k(t,x(h_k(t)))=0, $$ global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor-Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications., Comment: 20 pages, published in 2015 in Nonlinear Differential Equations and Applications

For three different types of angiogenesis models with variable delays, we consider either continuous or impulse therapy that eradicates tumor cells and suppresses angiogenesis. For the cancer-free solution, explicit conditions of global stability for the continuous and impulsive systems are obtained, together with delay-dependent estimates for the rates of decay for the tumor volume and pathological angiogenesis.

Theorems for the existence of periodic solutions for diverse models of population dynamics are obtained as corollaries of a few basic theorems, thus unifying the analysis of a broad class of scalar models in a single setting. The latter mechanism allows to obtain existence conditions for a broad class of nonlinear, non-autonomous models and models with state-dependent delays. The technique fulfills multiple roles: it can be used to expand on well-known results as well as to shorten existing proofs. We provide some examples which illustrate the applicability of our results.

a b s t r a c t Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.

To enrich the dynamics of mathematical models of angiogenesis, all mechanisms involved are time-dependent. We also assume that the tumor cells enter the mechanisms of angiogenic stimulation and inhibition with some delays. The models under study belong to a special class of nonlinear nonautonomous systems with delays. Explicit sufficient and necessary conditions for the existence of the positive periodic solutions were obtained via topological methods. Some open problems are presented for further studies. Fil: Amster, Pablo Gustavo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Berezansky, L.. Ben Gurion University Of The Negev; Israel Fil: Idels, L.. Vancouver Island University; Canadá

A class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques. The results for the nonautonomous systems with time-varying delays are novel, e.g., the global stability of differential equations with nonlinear (casual) Volterra operators is considered for the first time in the literature. Criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying/verifying in practice. We illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous Nicholson-type models.

Models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics that belong to the Nicholson-type delay differential systems are proposed. To study the global stability of the Nicholson-type models we construct an exponentially stable linear system such that its solution is a solution of the nonlinear model. Explicit conditions of the existence of positive global solutions, lower and upper estimations of solutions, and the existence and uniqueness of a positive equilibrium were obtained. New results, obtained for the global stability and instability of equilibria solutions, extend known results for the scalar Nicholson models. The conditions for the stability test are quite practical, and the methods developed are applicable to the modeling of a broad spectrum of biological processes. To illustrate our finding, we study the dynamics of the fish populations in Marine Protected Areas.

W e consider two regions with a fish population that is dispersing between the two areas, and fishing takes place only in region 2, with region 1 established as no-fishing zone. Marine protected areas (MPAs) have been promoted as conservation and fishery management tools, and at present, there are over 1300 MPAs in the world. A new mathematical model of an MPA that reflects the complexity of the natural setting is presented. The resulting model of an age-structured fish population belongs to a class of non-linear systems of differential equations with delay. New easily verifiable sufficient conditions for the existence, boundedness, permanence and stability of the positive internal steady-state solutions are obtained. From the point of view of fishery managers, the existence of stable solutions is necessary for planning harvesting strategies and sustaining the fishing grounds. Numerical simulations illustrate qualitative behaviour of the model, including stability switches.

This review covers permanence, oscillation, local and global stability of solutions for Nicholson’s blowflies differential equation. Some generalizations, including the most recent results for equations with a distributed delay and models with periodic coefficients, are considered.

For the equation, [email protected]?(t)=r(t)N(t)1+[N(t)]^@c-b(t)N(t)-a(t)N(g(t)), we obtain the following results: boundedness of all positive solutions, extinction, and persistence conditions. The proofs employ recent results in the theory of linear delay equations with positive and negative coefficients.

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points. Pre-print version This is a submitted manuscript of an article published by Taylor & Francis in International Journal of Control in 2015. The Version of Record of this manuscript has been published and is available at http://www.tandfonline.com/10.1080/00207179.2015.1009948. Berezansky, L., Braverman, E., & Idels, L. (2015). Stabilisation of second-order nonlinear equations with variable delay. International Journal of Control, 88(8), 1533-1539. DOI: 10.1080/00207179.2015.1009948 Pre-print version title has different spelling than final, published version title. https://viuspace.viu.ca/bitstream/handle/10613/3041/Idels.IJC.pdf?sequence=3

To enrich the dynamics of three models, we introduced biologically motivated time-varying delays. All models under study belong to a special class of nonlinear nonautonomous delay differential systems with non-Lipschitz nonlinearities. Explicit conditions for the existence of positive global solutions and the equilibria solutions were obtained. Based on a notion of an M-matrix, new results are presented for the global stability of the system and were used to prove local stability of one model. For a local stability of a second model, the recent result for a Lienard-type second-order differential equation with delays was used. It was shown that models with delays produce a complex and nontrivial dynamics. Some open problems are presented for further studies. Fil: Amster, Pablo Gustavo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Berezansky, L.. Ben Gurion University Of The Negev; Israel Fil: Idels, L.. Vancouver Island University; Canadá

In traditional harvesting model a fishing effort E is defined by the fishing intensity and does not address the inverse effect of fish abundance on the fishing effort. In this paper, based on a canonical differential equation model, we developed a new fishing effort model which relies on the density effect of fish population. We obtained new differential equations to describe certain standard Fisheries management strategies. This study concludes a control parameter(the magnitude of the effect of the fish population size on the fishing effort function E), changes not only the rate at which the population goes to equilibrium, but also the equilibrium values. To examine systematically the consequences of different harvesting strategies, we used numerical simulations and qualitative anlysis of six fishery strategies, e.g., proportional harvesting, threshold harvesting, proportional threshold harvesting, seasonal and rotational harvestinng., Comment: 16 pages and 20 figures