Pattern formation in a predator-mediated coexistence model with prey-taxis

Figure 1.  A plot of $ \chi $ versus $ k^{2} $. The red line is $ \chi_{c} $ as a function of $ k^{2} $. The blue dashed line shows $ a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi) = 0 $. Values of ($ k^{2}, \chi $) above (below) the blue dashed line correspond to $ a(k^{2})b(k^{2}, \chi)-c(k^{2}, \chi) $ positive (negative). This is an example where we use the parameter set: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, d) = (0.70, 0.70, 0.10, 0.30, 1.39, 0.30, 0.70, 0.25) $ Hence, $ a_{1}b_{2}-rb_{1} = 0.07 $, so $ k_{min}^{2} = 0.26 $

Figure 2.  A plot of $ \chi_{c_{1}} $ and $ \chi_{c_{2}} $ versus $ k^{2} $. $ \chi_{c_{1}} $ (red line) and $ \chi_{c_{2}} $ (blue dashed line) as functions of $ k^{2} $ are plotted for the parameter set: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu) = (0.10, 0.20, 0.10, 1.50, 3.0, 0.6, 0.1) $

Figure 3.  Case 1: $ u $ "wins": In this example the parameters are chosen so that in the pure competition system $ u $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted $ v $ is driven to extinction in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.25, 0.75, 0.75, 0.25, 0.6875, 0.5, 0.25, 15, 0.5) $

Figure 4.  Case 2: $ v $ "wins": In this example the parameters are chosen so that in the pure competition system $ v $ "wins". Note that while parameters are such that in the absence of a repulsive prey defense mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.1, 0.1, 1.4, 0.3, 0.4, 0.4, 0.2, 50, 0.25) $

Figure 5.  Spatio-temporal patterns: $ v $ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.0, 0.25, 0.5, 1.25, 0.5, 0.25, 25, 0.1) $

Figure 6.  Spatio-temporal patterns: $ u $ "wins" case. The first column shows the initial dynamics from the perturbation of the homogeneous steady state and the second column shows the spatio-temporal pattern that is achieved. In the last row $ u(x, t) $, $ v(x, t) $, and $ w(x, t) $ are given by the blue, green, and red lines respectively. The parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (0.9, 1.1, 0.2, 0.2, 1.09, 0.5, 1.0, 16, 0.25) $

Figure 7.  Attracting ($ \chi < 0 $) case: In this example the parameters are chosen so that in the pure competition system $ v $ "wins". Note that while parameters are such that in the absence of a attracing prey-taxis mechanism by $ u $, a predator mediated coexistence is expected; yet, once this state is disrupted, the prey coexist with peak densities out of phase with the predator in the presence of this taxis mechanism. Parameter set for this figure is: $ (a_{1}, a_{2}, b_{1}, b_{2}, r, \varepsilon, \mu, \chi, d) = (1.25, 1.00, 0.25, 0.50, 1.25, 0.5, 0.25, -0.1, 10) $