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Papers by Kelly Black

Research paper thumbnail of Long time dynamics of a three-species food chain model with Allee effect in the top predator

Computers & Mathematics with Applications, 2016

The Allee effect is an important phenomenon in population biology characterized by positive densi... more The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the the model, that is upper semicontinuous in the Allee threshold parameter m. To the best of our knowledge this is the first robustness result, for a spatially explicit three species food chain model with an Allee effect. Next, we numerically investigate the decay rate to a target attractor, that is when m = 0, in terms of m. We find decay estimates that are O(m γ ), where γ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m. Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.

Research paper thumbnail of Classroom Note: Putting Constraints in Optimization for First-Year Calculus Students

Siam Review, 1997

ABSTRACT

Research paper thumbnail of Complex Valued Spectral Hermite Approximations for the Actively Mode-Locked Laser

Journal of Scientific Computing, 2007

We construct a numerical approximation of the governing equations of an actively mode-locked lase... more We construct a numerical approximation of the governing equations of an actively mode-locked laser. The governing equation is complex valued and a novel scaling is employed that is designed to simplify the associated line integral in the complex plane. The resulting approximation is based on a set of shifted Hermite polynomials on an infinite line. Numerical comparisons are given with a finite difference scheme on a mapped domain as well as a finite element method on a truncated domain.

Research paper thumbnail of Extraction of Signals from Chaotic Laser Data

Physical Review Letters, 1999

Research paper thumbnail of Spectral Hermite Approximations for the Actively Mode-Locked Laser

Journal of Scientific Computing, 2001

An approximation technique for the governing equations for the mode-locked laser is examined. The... more An approximation technique for the governing equations for the mode-locked laser is examined. The technique centers on a transformation of the governing equations in which the resulting equations closely resemble the Hermite equation. The approximation of the system is constructed through a linear combination of Hermite polynomials resulting in a Hermite-spectral method. The rate of decay of the resulting modes is examined for a simplified problem and difficulties in showing the stability of the method are also discussed. Numerical comparisons with a finite difference scheme are also presented.

Research paper thumbnail of Extraction of Signals from Chaotic Laser Data

Physical Review Letters, 1999

Several experimental groups have demonstrated communication with chaotic lasers. We analyze data ... more Several experimental groups have demonstrated communication with chaotic lasers. We analyze data collected from a message-modulated erbium-doped fiber-ring laser (provided by VanWiggeren and Roy). We show that the transmitted signal is dominated by convolution of the message with the response function of the laser. A simple model based on the topology of the laser allows us to recover a hidden message. While prior estimates indicate that the laser dynamics are high dimensional, we show that only four parameters are required, each of which can be recovered from the transmitted signal alone.

Research paper thumbnail of Pulse Dynamics in an Actively Mode-Locked Laser

Siam Journal on Applied Dynamical Systems, 2003

We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitudem... more We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitudemodulated laser is subject to large transient growth and we demonstrate that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator. We also demonstrate an exact reduction from the governing PDEs to a low-dimensional system of ODEs for the parameters of an evolving pulse. A linearized version of these equations allows us to find analytical expressions for the transient growth below threshold. We also show that the nonlinear system collapses onto an appropriate fixed point, and thus in the absence of noise the ground-mode laser pulse is stable. We demonstrate numerically that, in the presence of a continuous noise source, however, the laser destabilizes and pulses are repeatedly created and annihilated.

Research paper thumbnail of Long time dynamics of a three-species food chain model with Allee effect in the top predator

Computers & Mathematics with Applications, 2016

The Allee effect is an important phenomenon in population biology characterized by positive densi... more The Allee effect is an important phenomenon in population biology characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We consider a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. We show the existence of a global attractor for the the model, that is upper semicontinuous in the Allee threshold parameter m. To the best of our knowledge this is the first robustness result, for a spatially explicit three species food chain model with an Allee effect. Next, we numerically investigate the decay rate to a target attractor, that is when m = 0, in terms of m. We find decay estimates that are O(m γ ), where γ is found explicitly. Furthermore, we prove various overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncover a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m. Our results have potential applications to trophic cascade control, conservation efforts in food chains, as well as Allee mediated biological control.

Research paper thumbnail of Classroom Note: Putting Constraints in Optimization for First-Year Calculus Students

Siam Review, 1997

ABSTRACT

Research paper thumbnail of Complex Valued Spectral Hermite Approximations for the Actively Mode-Locked Laser

Journal of Scientific Computing, 2007

We construct a numerical approximation of the governing equations of an actively mode-locked lase... more We construct a numerical approximation of the governing equations of an actively mode-locked laser. The governing equation is complex valued and a novel scaling is employed that is designed to simplify the associated line integral in the complex plane. The resulting approximation is based on a set of shifted Hermite polynomials on an infinite line. Numerical comparisons are given with a finite difference scheme on a mapped domain as well as a finite element method on a truncated domain.

Research paper thumbnail of Extraction of Signals from Chaotic Laser Data

Physical Review Letters, 1999

Research paper thumbnail of Spectral Hermite Approximations for the Actively Mode-Locked Laser

Journal of Scientific Computing, 2001

An approximation technique for the governing equations for the mode-locked laser is examined. The... more An approximation technique for the governing equations for the mode-locked laser is examined. The technique centers on a transformation of the governing equations in which the resulting equations closely resemble the Hermite equation. The approximation of the system is constructed through a linear combination of Hermite polynomials resulting in a Hermite-spectral method. The rate of decay of the resulting modes is examined for a simplified problem and difficulties in showing the stability of the method are also discussed. Numerical comparisons with a finite difference scheme are also presented.

Research paper thumbnail of Extraction of Signals from Chaotic Laser Data

Physical Review Letters, 1999

Several experimental groups have demonstrated communication with chaotic lasers. We analyze data ... more Several experimental groups have demonstrated communication with chaotic lasers. We analyze data collected from a message-modulated erbium-doped fiber-ring laser (provided by VanWiggeren and Roy). We show that the transmitted signal is dominated by convolution of the message with the response function of the laser. A simple model based on the topology of the laser allows us to recover a hidden message. While prior estimates indicate that the laser dynamics are high dimensional, we show that only four parameters are required, each of which can be recovered from the transmitted signal alone.

Research paper thumbnail of Pulse Dynamics in an Actively Mode-Locked Laser

Siam Journal on Applied Dynamical Systems, 2003

We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitudem... more We consider pulse formation dynamics in an actively mode-locked laser. We show that an amplitudemodulated laser is subject to large transient growth and we demonstrate that at threshold the transient growth is precisely the Petermann excess noise factor for a laser governed by a nonnormal operator. We also demonstrate an exact reduction from the governing PDEs to a low-dimensional system of ODEs for the parameters of an evolving pulse. A linearized version of these equations allows us to find analytical expressions for the transient growth below threshold. We also show that the nonlinear system collapses onto an appropriate fixed point, and thus in the absence of noise the ground-mode laser pulse is stable. We demonstrate numerically that, in the presence of a continuous noise source, however, the laser destabilizes and pulses are repeatedly created and annihilated.