It is illustrated that age-structured population models with an additional structure nicely fit i... more It is illustrated that age-structured population models with an additional structure nicely fit into the framework of Lipschitz perturbations of non-dense operators which satisfy the resolvent estimates. This makes it not only possible to show that unique solutions to the model equations exist in a well-defined sense, but also that they generate a dynamical system (or semiflow) which leaves closed convex subsets invariant that satisfy an appropriate subtangential condition. † supported by a Heisenberg scholarship of 'Deutsche Forschungsgemeinschaft'
A convolution of Stieltjes type is introduced for operator families of bounded strong variation a... more A convolution of Stieltjes type is introduced for operator families of bounded strong variation and vector valued L1-functions. Using this tool, perturbation theorems for integrated semigroups of bounded strong variation are derived, and improved results on Lp-solutions to the inhomogeneous Cauchy problem are obtained.
Transactions of the American Mathematical Society, 1995
From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an auto... more From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.
Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease... more Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, R 0 , exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep,R 0. If R 0 > 1 andR 0 > 1, bluetongue persists in cattle and midges even though it may eradicate the sheep, relying on cattle as a reservoir. If R 0 > 1 >R 0 , bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.
Convergence of almost all trajectories of strongly order-preserving semiflows is derived under su... more Convergence of almost all trajectories of strongly order-preserving semiflows is derived under suitable additional assumptions. These essentially consist in slightly sharpening the strongly order-preserving property and in the continuous differentiability of the flow with respect to the state variable. Required spectral properties of the linearizations of the flow around equilibria usually follow in the same way as the compactness and monotonicity assumptions for the flow itself. The proofs are based on sharpened versions of the limit set dichotomy and the sequential limit set trichotomy.
The global behavior of the general s s age-structured epidemic model in a population of constant ... more The global behavior of the general s s age-structured epidemic model in a population of constant size is obtained. It is shown that there is a sharp threshold which determines the existence and global stability of an endemic state; hence, periodic solutions are ruled out. The threshold is identified as the spectral radius of a positive linear operator. The analysis employs the theory of semigroups and positive operator methods, and is based on the formulation of the problem as an abstract differential equation in a Banach space. Key words, epidemic model, age structure, semigroup, monotone operator, global behavior AMS(MOS) subject classifications. 92A15, 35L60, 47D05, 47H07 *
An approach to persistence theory is presented which focuses on the concept of uniform weak persi... more An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.
The application of operator semigroups to Markov processes is extended to Markov transition funct... more The application of operator semigroups to Markov processes is extended to Markov transition functions which do not have the Feller property. Markov transition functions are characterized as solutions of forward and backward equations which involve the generators of integrated semigroups and are shown to induce integral semigroups on spaces of measures. Ω K(t, x, dy)f (y), f ∈ BM(Ω),
AbstractWe study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \small... more AbstractWe study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \smallint \limits_O^t \mathop \smallint \limits_D f(y, U (t - s,y)) h (x,y,s)dsdy,$$ t≥0, x∈D. We derive sufficient conditions for the boundedness of all non-negative solutions U. We show that, for bounded non-negative solutions U, U(t,.) is positive on D for sufficiently large t>0, if we impose appropriate positivity assumptions on f and h. If we additionally assume that, for y∈D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases, the following alternative holds for any bounded non-negative solution U: Either U(t,.) converges toward zero for t→∞, pointwise on D, or U(t,.) converges, for t→∞, toward the unique bounded positive solution of the corresponding Hammerstein integral equation, uniformly on D. We indicate conditions for the occurrence of each of the two cases.
By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) cond... more By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation $$\begin{array}{*{20}c} {u(x) = \int\limits_D {f(y,u(y)) k(x,y) dy ,} } & {x \in D,} \\ \end{array} $$ and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem.
Journal of Mathematical Analysis and Applications, 1990
By introducing a stronger than pointwise ordering, conditions are found under which scalar functi... more By introducing a stronger than pointwise ordering, conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi-monotone. Typically the maximum delay must be the smaller the more quasi-monotonicity is violated. The theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations.
Journal of Mathematical Analysis and Applications, 1990
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Journal of Mathematical Analysis and Applications, 1998
In this paper, the strong and uniform approach to balanced exponential growth is characterized an... more In this paper, the strong and uniform approach to balanced exponential growth is characterized and applicable sufficient conditions are derived. The results will be applied to models for age-size-structured population dynamics in a forthcoming publication.
Journal of Dynamics and Differential Equations, 2003
In the present paper we consider the nonlinear evolution equation uOE+Au ¦ G(u), where A: D(A) ı ... more In the present paper we consider the nonlinear evolution equation uOE+Au ¦ G(u), where A: D(A) ı X Q X is m-accretive with (I+lA) −1 compact for some l > 0, and G: D(A) Q X is continuous, and we prove that the orbit {u(t); t ¥ R + } is relatively compact if and only if u is uniformly continuous, and both u and G p u are bounded on R +. In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.
It is illustrated that age-structured population models with an additional structure nicely fit i... more It is illustrated that age-structured population models with an additional structure nicely fit into the framework of Lipschitz perturbations of non-dense operators which satisfy the resolvent estimates. This makes it not only possible to show that unique solutions to the model equations exist in a well-defined sense, but also that they generate a dynamical system (or semiflow) which leaves closed convex subsets invariant that satisfy an appropriate subtangential condition. † supported by a Heisenberg scholarship of 'Deutsche Forschungsgemeinschaft'
A convolution of Stieltjes type is introduced for operator families of bounded strong variation a... more A convolution of Stieltjes type is introduced for operator families of bounded strong variation and vector valued L1-functions. Using this tool, perturbation theorems for integrated semigroups of bounded strong variation are derived, and improved results on Lp-solutions to the inhomogeneous Cauchy problem are obtained.
Transactions of the American Mathematical Society, 1995
From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an auto... more From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.
Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease... more Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, R 0 , exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep,R 0. If R 0 > 1 andR 0 > 1, bluetongue persists in cattle and midges even though it may eradicate the sheep, relying on cattle as a reservoir. If R 0 > 1 >R 0 , bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.
Convergence of almost all trajectories of strongly order-preserving semiflows is derived under su... more Convergence of almost all trajectories of strongly order-preserving semiflows is derived under suitable additional assumptions. These essentially consist in slightly sharpening the strongly order-preserving property and in the continuous differentiability of the flow with respect to the state variable. Required spectral properties of the linearizations of the flow around equilibria usually follow in the same way as the compactness and monotonicity assumptions for the flow itself. The proofs are based on sharpened versions of the limit set dichotomy and the sequential limit set trichotomy.
The global behavior of the general s s age-structured epidemic model in a population of constant ... more The global behavior of the general s s age-structured epidemic model in a population of constant size is obtained. It is shown that there is a sharp threshold which determines the existence and global stability of an endemic state; hence, periodic solutions are ruled out. The threshold is identified as the spectral radius of a positive linear operator. The analysis employs the theory of semigroups and positive operator methods, and is based on the formulation of the problem as an abstract differential equation in a Banach space. Key words, epidemic model, age structure, semigroup, monotone operator, global behavior AMS(MOS) subject classifications. 92A15, 35L60, 47D05, 47H07 *
An approach to persistence theory is presented which focuses on the concept of uniform weak persi... more An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.
The application of operator semigroups to Markov processes is extended to Markov transition funct... more The application of operator semigroups to Markov processes is extended to Markov transition functions which do not have the Feller property. Markov transition functions are characterized as solutions of forward and backward equations which involve the generators of integrated semigroups and are shown to induce integral semigroups on spaces of measures. Ω K(t, x, dy)f (y), f ∈ BM(Ω),
AbstractWe study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \small... more AbstractWe study the Volterra-Hammerstein integral equation $$U(t,x) = U_O (t,x) + \mathop \smallint \limits_O^t \mathop \smallint \limits_D f(y, U (t - s,y)) h (x,y,s)dsdy,$$ t≥0, x∈D. We derive sufficient conditions for the boundedness of all non-negative solutions U. We show that, for bounded non-negative solutions U, U(t,.) is positive on D for sufficiently large t>0, if we impose appropriate positivity assumptions on f and h. If we additionally assume that, for y∈D, rf(y,r) strictly monotone increases and f(y,r)/r strictly monotone decreases as r>0 increases, the following alternative holds for any bounded non-negative solution U: Either U(t,.) converges toward zero for t→∞, pointwise on D, or U(t,.) converges, for t→∞, toward the unique bounded positive solution of the corresponding Hammerstein integral equation, uniformly on D. We indicate conditions for the occurrence of each of the two cases.
By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) cond... more By a monotone representation of the nonlinearity we derive sufficient (and partly necessary) conditions for the unique existence of positive solutions of the Hammerstein integral equation $$\begin{array}{*{20}c} {u(x) = \int\limits_D {f(y,u(y)) k(x,y) dy ,} } & {x \in D,} \\ \end{array} $$ and for the convergence of successive approximations towards the solution. Further we study the corresponding nonlinear eigenvalue problem.
Journal of Mathematical Analysis and Applications, 1990
By introducing a stronger than pointwise ordering, conditions are found under which scalar functi... more By introducing a stronger than pointwise ordering, conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi-monotone. Typically the maximum delay must be the smaller the more quasi-monotonicity is violated. The theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations.
Journal of Mathematical Analysis and Applications, 1990
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
Journal of Mathematical Analysis and Applications, 1998
In this paper, the strong and uniform approach to balanced exponential growth is characterized an... more In this paper, the strong and uniform approach to balanced exponential growth is characterized and applicable sufficient conditions are derived. The results will be applied to models for age-size-structured population dynamics in a forthcoming publication.
Journal of Dynamics and Differential Equations, 2003
In the present paper we consider the nonlinear evolution equation uOE+Au ¦ G(u), where A: D(A) ı ... more In the present paper we consider the nonlinear evolution equation uOE+Au ¦ G(u), where A: D(A) ı X Q X is m-accretive with (I+lA) −1 compact for some l > 0, and G: D(A) Q X is continuous, and we prove that the orbit {u(t); t ¥ R + } is relatively compact if and only if u is uniformly continuous, and both u and G p u are bounded on R +. In the same spirit, we derive conditions for orbits of bounded sets to have compact attractors. Some consequences and an example from age-structured population dynamics illustrate the effectiveness of the abstract result.
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