 Award ID(s):
 2006887
 Publication Date:
 NSFPAR ID:
 10335776
 Journal Name:
 IMA Journal of Applied Mathematics
 Volume:
 87
 Issue:
 2
 Page Range or eLocationID:
 131 to 186
 ISSN:
 02724960
 Sponsoring Org:
 National Science Foundation
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Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopfbifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Timemore »

We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thinlayer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The threecurve equations of contour dynamics that fully describe the doublelayer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initialvalue evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the singlelayer equation of Moore ( Stud. Appl. Math. , vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the firstorder equations to smallamplitude perturbations shows Kelvin–Helmholtz instability when the farfield fluid velocities on either side of the double layer are unequal. Equal velocities define a circulationfree double vorticity layer,more »

Recent theoretical and computational progress has led to unprecedented understanding of symmetrybreaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale ℓ and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wavespeed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with ℓ, and at larger loading levels (corresponding to yet higher propagation velocities), a tipsplitting instability emerges, both in agreements with experiments. In this paper, using phasefield models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of neartip elastic nonlinearity, i.e. in the limit ℓ→0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii)more »

In this paper, an insectparasitehost model with logistic growth of triatomine bugs is formulated to study the transmission between hosts and vectors of the Chagas disease by using dynamical system approach. We derive the basic reproduction numbers for triatomine bugs and
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