arXiv subjects

Erika Gallo

Publications and source records attributed to Erika Gallo.

4 recordsLinked to original sources

arXiv2026

A Model of Annual Tick Population Density in the Eastern United States as a Function of Questing Behavior and Host Availability

Understanding the ecological processes that regulate populations of the blacklegged tick, Ixodes scapularis, is essential for explaining regional differences in tick abundance and the risk of tick-borne diseases. In this study, we develop a stage-structured nonlinear system of difference equations by ordering and composing the seasonal biological events that govern tick and host life cycles. The model incorporates ratio-dependent host-attachment, allowing host-finding success to depend jointly on host availability and tick abundance. Parameter estimates from both northeastern and southeastern ecosystems in the United States are used to examine how regional differences in host community composition and questing behavior influence tick population dynamics. Analysis shows that a $+1$/$-1$ bifurcation pair occurs at the demographic persistence threshold, producing an exchange of stability between the extinction and positive fixed points together with the emergence of an unstable cohort-based 2-cycle. Quantitative results show that ratio-dependent host-finding success limits population growth by restricting successful feeding, while questing behavior alters both the frequency and composition of host encounters. Together, these mechanisms regulate tick persistence and long-term abundance, providing a demographic foundation for understanding variation in tick abundance and Lyme disease risk.

math.DS↗

arXiv2026

Stability of Ginzburg-Landau pulses via Fredholm determinants of Birman-Schwinger operators

We introduce a numerical method to determine the stability of stationary pulse solutions of the complex Ginzburg-Landau equation. The method involves the computation of the point spectrum of the first-order linear differential operator with matrix-valued coefficients on the real line obtained by linearizing the Ginzburg-Landau equation about a stationary pulse. Applying a general theory of Gesztesy, Latushkin, and Makarov, we show that this point spectrum is given by the set of zeros of a 2-modified Fredholm determinant of a Hilbert-Schmidt, Birman-Schwinger operator. We establish conditions which guarantee that this operator is trace class. Applying results of Bornemann on the numerical computation of Fredholm determinants, we obtain a bound on the error between the regular Fredholm determinant of the trace class operator and its numerical approximation by a matrix determinant. We verify the new numerical Fredholm determinant method for computing the point spectrum of a Ginzburg-Landau pulse by exhibiting excellent agreement with previous methods. This new approach avoids the challenge of solving the numerically stiff system of equations for the matrix-valued Jost solutions, and it opens the way for the spectral analysis of breather solutions of nonlinear wave equations, for which an Evans function does not exist.

math.SP↗

arXiv2025

Numerical Fredholm determinants for matrix-valued kernels on the real line

We analyze a numerical method for computing Fredholm determinants of trace class and Hilbert Schmidt integral operators defined in terms of matrix-valued kernels on the entire real line. With this method, the Fredholm determinant is approximated by the determinant of a matrix constructed by truncating the kernel of the operator to a finite interval and then applying a quadrature rule. Under the assumption that the kernel decays exponentially, we derive an estimate relating the Fredholm determinant of the operator on the real line to that of its truncation to a finite interval. Then we derive a quadrature error estimate relating the Fredholm determinant of a matrix-valued kernel on a finite interval to its numerical approximation obtained via an adaptive composite Simpson's quadrature rule. These results extend the analysis of Bornemann which focused on Fredholm determinants of trace class operators defined by scalar-valued kernels on a finite interval. Numerical results are provided for a Birman-Schwinger operator that characterizes the stability of stationary solutions of nonlinear wave equations.

math.NA↗

arXiv2024

A regularity condition under which integral operators with operator-valued kernels are trace class

We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $\dim H < \infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.

math.FA↗