On stability of cooperative and hereditary systems with a distributed delay

We consider a system $\displaystyle \frac{dx}{dt}=r_1(t) G_1(x) \left[ \int_{h_1(t)}^t f_1(y(s))~d_s R_1 (t,s) - x(t) \right], \frac{dy}{dt}=r_2(t) G_2(y) \left[ \int_{h_2(t)}^t f_2(x(s))~d_s R_2 (t,s) - y(t)\right]$ with increasing functions $f_1$ and $f_2$, which has at most one positive equilibrium. Here the values of the functions $r_i,G_i,f_i$ are positive for positive arguments, the delays in the cooperative term can be distributed and unbounded, both systems with concentrated delays and integro-differential systems are a particular case of the considered system. Analyzing the relation of the functions $f_1$ and $f_2$, we obtain several possible scenarios of the global behaviour. They include the cases when all nontrivial positive solutions tend to the same attractor which can be the positive equilibrium, the origin or infinity. Another possibility is the dependency of asymptotics on the initial conditions: either solutions with large enough initial values tend to the equilibrium, while others tend to zero, or solutions with small enough initial values tend to the equilibrium, while others infinitely grow. In some sense solutions of the equation are intrinsically non-oscillatory: if both initial functions are less/greater than the equilibrium value, so is the solution for any positive time value. The paper continues the study of equations with monotone production functions initiated in [Nonlinearity, 2013, 2833-2849].Comment: 18 pages, 3 figures, published in 2015 in Nonlinearit

On stability of delay equations with positive and negative coefficients with applications

We obtain new explicit exponential stability conditions for linear scalar equations with positive and negative delayed terms $$\dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))- \sum_{k=1}^l b_k(t)x(g_k(t))=0$$ and its modifications, and apply them to investigate local stability of Mackey--Glass type models $$\dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(g(t))}-\gamma x(h(t))\right]$$ and $$\dot{x}(t)=r(t)\left[\beta\frac{x(g(t))}{1+x^n(h(t))}-\gamma x(t)\right].$$Comment: 33 pages, 3 figure

Stabilization of difference equations with noisy proportional feedback control

Given a deterministic difference equation $x_{n+1}= f(x_n)$, we would like to stabilize any point $x^{\ast}\in (0, f(b))$, where $b$ is a unique maximum point of $f$, by introducing proportional feedback (PF) control. We assume that PF control contains either a multiplicative $x_{n+1}= f\left( (\nu + \ell\chi_{n+1})x_n \right)$ or an additive noise $x_{n+1}=f(\lambda x_n) +\ell\chi_{n+1}$. We study conditions under which the solution eventually enters some interval, treated as a stochastic (blurred) equilibrium. In addition, we prove that, for each $\varepsilon>0$, when the noise level $\ell$ is sufficiently small, all solutions eventually belong to the interval $(x^{\ast}-\varepsilon,x^{\ast}+\varepsilon)$.Comment: 20 pages, 19 figure

On the interplay of harvesting and various diffusion strategies for spatially heterogeneous populations

The paper explores the influence of harvesting (or culling) on the outcome of the competition of two species in a spatially heterogeneous environment. The harvesting effort is assumed to be proportional to the space-dependent intrinsic growth rate. The differences between the two populations are the diffusion strategy and the harvesting intensity. In the absence of harvesting, competing populations may either coexist, or one of them may bring the other to extinction. If the latter is the case, introduction of any level of harvesting to the successful species guarantees survival to its non-harvested competitor. In the former case, there is a strip of "close enough" to each other harvesting rates leading to preservation of the original coexistence. Some estimates are obtained for the relation of the harvesting levels providing either coexistence or competitive exclusion.Comment: 25 pages, 11 figure

Boundedness and persistence of delay differential equations with mixed nonlinearity

For a nonlinear equation with several variable delays $$\dot{x}(t)=\sum_{k=1}^m f_k(t, x(h_1(t)),\dots,x(h_l(t)))-g(t,x(t)),$$ where the functions $f_k$ increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on $[0, \infty)$, as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.Comment: 24 pages, published in Applied Mathematics and Computation, 201

On stability of linear neutral differential equations with variable delays

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0,$$ where $$|a(t)|<1,~ b(t)\geq 0, ~h(t)\leq t, ~g(t)\leq t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.Comment: 28 pages. to appear in Czechoslovak Mathematical Journal, published onlin

On convergence of solutions to difference equations with additive perturbations

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behavior of the map function $f$ at zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the map $f$ at zero exceeds one or, independently of the behavior of $f$ at zero, when the amplitudes are not square summable.Comment: 22 pages, 4 figures, to appear in Journal of Difference Equations and Application

Stochastic difference equations with the Allee effect

For a truncated stochastically perturbed equation $x_{n+1}=\max\{ f(x_n)+l\chi_{n+1}, 0 \}$ with $f(x)<x$ on $(0,m)$, which corresponds to the Allee effect, we observe that for very small perturbation amplitude $l$, the eventual behavior is similar to a non-perturbed case: there is extinction for small initial values in $(0,m-\varepsilon)$ and persistence for $x_0 \in (m+\delta, H]$ for some $H$ satisfying $H>f(H)>m$. As the amplitude grows, an interval $(m-\varepsilon, m+\delta)$ of initial values arises and expands, such that with a certain probability, $x_n$ sustains in $[m, H]$, and possibly eventually gets into the interval $(0,m-\varepsilon)$, with a positive probability. Lower estimates for these probabilities are presented. If $H$ is large enough, as the amplitude of perturbations grows, the Allee effect disappears: a solution persists for any positive initial value.Comment: 17 pages, 15 figures, to appear in Dynamics of Continuous and Discrete Systems - Series

Stability conditions for scalar delay differential equations with a nondelay term

The problem considered in the paper is exponential stability of linear equations and global attractivity of nonlinear non-autonomous equations which include a non-delay term and one or more delayed terms. First, we demonstrate that introducing a non-delay term with a non-negative coefficient can destroy stability of the delay equation. Next, sufficient exponential stability conditions for linear equations with concentrated or distributed delays and global attractivity conditions for nonlinear equations are obtained. The nonlinear results are applied tothe Mackey-Glass model of respiratory dynamics.Comment: 12 pages, 2 figures, published in 2015 in Applied Mathematics and Computatio

Structured stability radii and exponential stability tests for Volterra difference systems

Uniform exponential (UE) stability of linear difference equations with infinite delay is studied using the notions of a stability radius and a phase space. The state space \X is supposed to be an abstract Banach space. We work both with non-fading phase spaces c_0 (\ZZ^-, \X) and \ll^\infty (\ZZ^-, \X) and with exponentially fading phase spaces of the $\ll^p$ and $c_0$ types. For equations of the convolution type, several criteria of UE stability are obtained in terms of the Z-transform \wh K (\zeta) of the convolution kernel $K (\cdot)$, in terms of the input-state operator and of the resolvent (fundamental) matrix. These criteria do not impose additional positivity or compactness assumptions on coefficients $K(j)$. Time-varying (non-convolution) difference equations are studied via structured UE stability radii \r_\t of convolution equations. These radii correspond to a feedback scheme with delayed output and time-varying disturbances. We also consider stability radii \r_\c associated with a time-invariant disturbance operator, unstructured stability radii, and stability radii corresponding to delayed feedback. For all these types of stability radii two-sided estimates are obtained. The estimates from above are given in terms of the Z-transform \wh K (\zeta), the estimate from below via the norm of the input-output operator. These estimates turn into explicit formulae if the state space \X is Hilbert or if disturbances are time-invariant. The results on stability radii are applied to obtain various exponential stability tests for non-convolution equations. Several examples are provided.Comment: Submitted to "Computers and Mathematics with Applications" (a special issue of proceedings of the conference "Progress in Difference Equations - 2012, Richmond, VA, May 13-18, 2012