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Figure 1.
Typical examples of the function $ f(u) $. It can be monotone (left) or non-monotone (right)
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Figure 2.
Period doubling bifurcations for the non-monotone function $ f(u) $. Upper row: simple oscillations ($ f_3 = 2.5 $) and period 2 oscillations ($ f_3 = 2 $). Lower row: period 4 oscillations ($ f_3 = 1.899 $) and period 8 oscillations ($ f_3 = 1.898 $). The dots show the beginning of the periods. The values of parameters: $ f_1 = 0, f_2 = 0.1, f_3 $ varies, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 2 $
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Figure 3.
Behavior of solutions of equation (1) with periodic boundary conditions for $ D = 0.001 $ (left), $ D = 0.0004 $ (middle), $ D = 0.0001 $ (right). The upper row of the graphs specifies the spatial pattern of solutions at some time $ t $. The lower row shows the spatiotemporal structure of solutions with the positions of the maxima of solutions in time. The values of parameters: $ f_1 = 0.1, f_2 = 6, f_3 = 6 $, $ u_1 = 0.1, u_2 = 0.9, u_3 = 0.95 $, $ \tau = 1.25 $, $ L = 0.5 $
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Figure 4.
Behavior of solutions of equation (1) with periodic boundary conditions for $ D = 10^{-3} $ (left), $ D = 10^{-4} $ (middle), $ D = 10^{-5} $ (right). The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 = 1.9 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 2 $, $ L = 0.5 $
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Figure 5.
Evolution of the transition zone in time. Upper row: time oscillations at the center of the interval. Middle row: $ u(x, t) $ as a function of $ x $ for some fixed $ t $. Lower row: positions of the maxima of solutions. The values of parameters: $ f_1 = 0.1, f_2 = 3, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.9, u_3 = 0.95 $, $ \tau = 1.5 $ (left); $ f_1 = 0.1, f_2 = 3, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.9, u_3 = 0.95 $, $ \tau = 2 $ (middle); $ f_1 = 0.1, f_2 = 6, f_3 = 6 $, $ u_1 = 0.1, u_2 = 0.9, u_3 = 0.95 $, $ \tau = 1.5 $ (right). $ u_0 = 0.2, u_1^0 = 0.001 $, $ D = 10^{-5}, L = 1 $
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Figure 6.
Quasi-waves for the non-monotone function $ f(u) $, $ f_3 = 3 $ (left); $ f_3 = 2.3 $ (middle); $ f_3 = 1.9 $ (right). The figure shows snapshots of solutions (upper row) and location of the maxima of solutions on the $ (x, t) $-plane (lower row). The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 $ varied, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 2 $, $ u_0 = 0.2, u_1^0 = 0.001 $
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Figure 7.
Quasi-waves for the non-monotone function $ f(u) $, $ \tau = 5 $ (left); $ \tau = 7 $ (right). Middle column shows the corresponding homogeneous oscillations for $ \tau = 5 $ (upper) and $ \tau = 7 $ (lower). The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 = 2 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau $ varied, $ u_0 = 0.2, u_1^0 = 0.001 $
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Figure 8.
Three types of regimes: wave with the minimal speed $ c_0 $ and a constant value $ v_0 $ behind the wave (left); the same wave is followed periodic in time and constant in space oscillations (middle); the wave converges to a periodic space structure instead of the constant value (right). The wave and the interval of time periodic oscillations are separated by a transition zone growing linearly in time. The values of parameters: $ u_1 = 0.1, u_2 = 0.9, u_3 = 0.95 $, $ \tau = 2 $, $ u_0 = 0.2, u_1^0 = 0 $; $ f_1 = 0.1, f_2 = 0.9, f_3 = 0.9 $ (left); $ f_1 = 0.1, f_2 = 1.6, f_3 = 1.6 $ (middle), $ f_1 = 0.1, f_2 = 3, f_3 = 3 $ (right)
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Figure 9.
Different regimes in the case of non-monotone function $ f(u) $. In all three cases, the most left is the bistable wave, next to it a periodic wave, the most right is time periodic homogeneous in space solution. The speed of the bistable wave is negative in the left image while the speed of the periodic wave is positive. The speed of the bistable wave is larger than the speed of the periodic wave in the middle image. As a consequence, they merge forming a single periodic bistable wave. The periodic wave is separated from the homogeneous solutions by a quasi-wave in the right image. The values of parameters: $ f_1 = 0, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.5, u_3 = 0.9 $, $ \tau = 2 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (left); $ f_1 = 0, f_2 = 0.1, f_3 = 1.2 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 1.2 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (middle); $ f_1 = 0, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 2 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (right)
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Figure 10.
Different regimes in the case of non-monotone function $ f(u) $. In all three cases, bistable waves are separated from the homogeneous solutions by quasi-waves with different structures. The values of parameters: $ f_1 = 0, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.5, u_3 = 0.9 $, $ \tau = 4 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (left); $ f_1 = 0, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 6 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (middle); $ f_1 = 0.1, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 1 $, $ u_0 = 0.8, u_1^0 = 0.001 $ (right)
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Figure 11.
The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 = 1.6 $, $ u_1 = 0.1, u_2 = 0.5, u_3 = 0.9 $, $ \tau = 1 $, $ u_0 = 0.8, u_1^0 = 0 $ (left); $ f_1 = 0, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 6 $, $ u_0 = 0.8, u_1^0 = 0 $ (middle); $ f_1 = 0.1, f_2 = 0.1, f_3 = 3 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ \tau = 1 $, $ u_0 = 0.8, u_1^0 = 0 $ (right)
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Figure 12.
Numerical simulation of equation (1) in a square domain with no-flux boundary condition and radially symmetric initial condition. The circular waves propagate from the lower left corner and gradually fill the whole domain. The left image shows a snapshot of solution, the right image the level lines of the same solution in a color code. The middle image shows the maxima of solution in time. Each green surface corresponds to the line of maxima (wave contour) propagating in space. The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 = 2 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.6 $, $ \tau = 2 $, $ u_0 = 0.2, u_1^0 = 0.001 $, $ D = 0.001 $
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Figure 13.
Snapshot of solutions with quasi-wave propagation for $ \tau = 2 $ (left) and $ \tau = 4 $ (right). The values of parameters: $ f_1 = 0.1, f_2 = 0.1, f_3 = 1.8 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.5 $, $ u_0 = 0.2, u_1^0 = 0.001 $, $ D = 0.0001 $
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Figure 14.
Propagation of a monostable-bistable wave. The initial condition (upper left) has a 2D perturbation. The wave remain 1D (lower left). When a quasi-wave develops between the bistable and the periodic monostable waves, the oscillations become two-dimensional (upper and lower right). The values of parameters: $ f_1 = 0.2, f_2 = 0.1, f_3 = 1.5 $, $ u_1 = 0.1, u_2 = 0.3, u_3 = 0.6 $, $ \tau = 4 $, $ D = 0.0002 $
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