Amplitude mediated spiral chimera pattern in a nonlinear reaction-diffusion system

The equilibrium point $(u_{*},v_{*})$ of the system (1) in the absence of diffusion (i.e., $D_{u}=D_{v}=0$) is given by the solution of the equations $f(u_{*},v_{*})=0$ and $g(u_{*},v_{*})=0$, where

Solving Eq. (3) we can easily detect the following different possible equilibrium points of the system:
(i) Extinction of both the prey and predator population: $E_{00}=(0,0)$.
(ii) Existence of only prey population: $E_{u0}=(k,0)$, where the prey population varies depending on the carrying capacity $k$.
(iii) Coexistence of both the populations: $E_{uv}=(u_{s},v_{s})$, where $v_{s}=\frac{1}{\eta}\Big{(}\frac{cu_{s}^{2}}{1+u_{s}^{2}}-d\Big{)}$ and $u_{s}$ is a positive root of the following quintic equation:

Based on the parametric constraints, the number of interior equilibrium points (coexistence of both the prey and predator populations) can vary from zero to five. Here we plot the number of feasible co-existence equilibrium points $(u_{s},v_{s})$ in different parameter spaces as depicted in Fig. 1. The following parameter values $k=6.5,b=0.1923,c=0.8,d=0.5$ epjst_connect are fixed in such a manner that the system (1) exhibits oscillatory dynamics in the absence of diffusion with proper choice of the parameters $a$ and $\eta$. Looking into the $k-\eta$ parameter space plotted in Fig. 1(a), we fix the value of intra-predator interaction rate $\eta$ at $\eta=0.018$ for the remaining numerical simulations. In the figure, the red region corresponds to the existence of only one interior equilibrium point while the black region signifies the absence of coexistence solution (i.e., negative real roots of Eq. (4)). Thus the parameter spaces depicted in all the figures in Fig. 1 indicate that the system (1) has maximally one interior equilibrium point in the absence of diffusion for our chosen values of the parameters.

The eigenvalue $\lambda$ of the Jacobian matrix $J$ of the linearized system determining the stability book_ml of the coexistence equilibrium $(u_{s},v_{s})$, satisfies the equation $\lambda^{2}-Tr(J)\lambda+\Delta_{J}=0$. Thus,

where $Tr(J)$ and $\Delta_{J}$ are the trace and determinant of the Jacobian matrix given by

In Eq. (6) $f_{u},f_{v},g_{u},g_{v}$ are the partial derivatives of the functions $f,g$ with respect to $u,v$, respectively. The equilibrium point is stable if $Tr(J)$ and $\Delta_{J}$ evaluated at $(u_{s},v_{s})$ necessarily satisfy the conditions $Tr(J)<0$, $\Delta_{J}>0$, and unstable otherwise. The system exhibits a Hopf bifurcation when the eigenvalues cross the imaginary axis at a point where $Tr(J)=0$, and the threshold for the bifurcation is derived as $a_{0}=-\frac{g_{v}}{f_{u}}$.

Now we investigate the condition for Turing instability of the spatiotemporal system (1) where the spatially homogeneous stable steady state $(u_{s},v_{s})$ becomes unstable to small amplitude perturbations in the presence of diffusion coefficients $D_{u}$ and $D_{v}$. Linearizing the system around the interior equilibrium point $(u_{s},v_{s})$, we can write

where $(\tilde{u},\tilde{v})$ is the small perturbation around $(u_{s},v_{s})$ such that $u=u_{s}+\tilde{u}$ and $v=v_{s}+\tilde{v}$. We consider,

as the solution of the linearized system $\eqref{eq7}$. Here, $\lambda$ determines the rate of evolution of the perturbation over time $t$, $\vec{\sigma}=(\sigma_{x},\sigma_{y})$ is the wave number such that $\vec{\sigma}.\vec{\sigma}=\sigma^{2}$ and $\vec{r}=(x,y)$ is the spatial position in two dimensions. In (8), $\tilde{u}_{\sigma},\tilde{v}_{\sigma}$ are the amplitudes of the perturbation. The Jacobian $J_{D}$ of the linearized system (7) is therefore given by

Hence, the characteristic equation can be written as

where $Tr(J_{D})=Tr(J)-(D_{u}+D_{v})\sigma^{2}$ and $\Delta_{J_{D}}=D_{u}D_{v}\sigma^{4}-(D_{u}g_{v}+aD_{v}f_{u})\sigma^{2}+\Delta_{J}$. For the emergence of Turing instability, $(u_{s},v_{s})$ must be stable in the absence of diffusion which guarantees that $Tr(J)<0$ and $\Delta_{J}>0$. Therefore, from the expression of $Tr(J_{D})$, it is clear that $Tr(J_{D})<0$. Obviously, in order to induce instability in the spatiotemporal model (1) in the presence of diffusion coefficients $D_{u}$ and $D_{v}$, $\Delta_{J_{D}}$ must be negative, i.e., $D_{u}D_{v}\sigma^{4}-(D_{u}g_{v}+aD_{v}f_{u})\sigma^{2}+\Delta_{J}<0$. But, the diffusion coefficients $D_{u},D_{v}$ as well as $\Delta_{J}$ all are positive, so necessarily the coefficient of $\sigma^{2}$ in the expression of $\Delta_{J_{D}}$ should be positive to generate Turing instability. Thus,

is the necessary condition for the instability of the equilibrium point $(u_{s},v_{s})$ in the presence of diffusion. Combining the condition (11) and $Tr(J)<0$, it eventually implies that $D_{v}>D_{u}$ (since $f_{u}(u_{s},v_{s})>0$ and $g_{v}(u_{s},v_{s})<0$) for the equilibrium to be unstable.

Now, we consider the expression

which is a quadratic equation in $p$ with $p=\sigma^{2}$. Here we plot the quadratic expression $h(p)$ with respect to $p$ in Fig. 2(a) for different values of the parameter $a$. It is easily discernible from the figure that as the value of $a$ increases the curve of $h(p)$ moves downwards and thereby the width of the interval for which $h(p)<0$ increases. So, it is evident that the possibility of occurrence of Turing instability increases as the system parameter $a$ is increased. For better visibility in Fig. 2(b), we show the zoomed version of the red rectangular region near $p=1$ depicted in Fig. 2(a). This figure confirms that the determinant $h(0)$ of the Jacobian of the non-spatial system is positive for all considered values of $a$. The graph of $h(p)$ with respect to $p$ represents a parabolic structure for which the minimum is reached at $p=p_{m}$, following the relation $\frac{dh(p)}{dp}=0$. Solving this we obtain the threshold of $p$ as $p_{m}=\frac{D_{u}g_{v}+aD_{v}f_{u}}{2D_{u}D_{v}}$ at which the function $h(p)$ attains its minima, as $\frac{d^{2}h(p)}{dp^{2}}>0$, and the minimum value is given by

Using the condition (13) we can derive the expression for the critical value of diffusion coefficient $D_{v}$ beyond which Turing instability emerges in the spatiotemporal system and this value is obtained as

Figure 3 depicts the critical value (14) in red for a wide range of the parameter values of $a$. Here the blue dashed vertical line corresponds to the Hopf bifurcation threshold $a_{0}$. These two Hopf and Turing instability curves intersect at a particular point and divide the whole parametric space into four distinct domains. Below the Turing curve in the domain I, the system reaches the spatially homogeneous stationary state $(u_{s},v_{s})$, while in domain II above the Turing curve the spatially homogeneous steady state loses its stability and the system reaches inhomogeneous steady states which confirms the appearance of Turing patterns. On the other side, beyond the Hopf boundary in domain III, Hopf instability occurs and the system exhibits limit cycle oscillations, whereas the most interesting non-trivial dynamics is observed in domain IV where both Hopf and Turing instabilities take place. Figure 4 depicts the spatial patterns of the prey species for some exemplary parameter values from the regions I, II and III. To generate the spatial patterns, the initial distribution of the species is considered as

where $(x,y)$ is considered as the grid index and $\epsilon_{1},\epsilon_{2}$ are small fluctuations added to the homogeneous steady state. We take $\epsilon_{1}=\epsilon_{2}=10^{-4}$. The appearance of spatially homogeneous steady state in region I is portrayed in Fig. 4(a) for the parameter value $a=1.3$ and diffusion strengths $D_{u}=0.01,D_{v}=0.3$. Figures 4(b) and 4(c) correspond to the Turing patterns when the parameter values are chosen from region II. Stripe pattern is depicted in Fig. 4(b) for the parameter values $a=1.3,D_{u}=0.01$ and $D_{v}=0.8$ while a combination of stripes and spots is detected for the parameter values $a=1.3,D_{u}=0.01$ and $D_{v}=1.2$ in Fig. 4(c). An intriguing spiral pattern is illustrated in Fig. 4(d) for the parameter values $a=4.0,D_{u}=0.01,D_{v}=0.01$ set in region III. In the following section, we will explore the rich dynamical patterns observed on the right of the Hopf boundary below and above the Turing curve.

From a detailed numerical analysis, we find that three different spatial patterns arise in the $D_{u}-D_{v}$ parameter space near the Hopf boundary for the parameter value $a=2.0$ as presented in Fig. 5. The regime diagram is obtained by looking into the snapshot at the corresponding diffusion parameter values at the final time when the system is simulated up to $t=5\times 10^{4}$ time units. In addition, we also checked the temporal as well as the spatiotemporal dynamics in the time interval $[4.99\times 10^{4},5\times 10^{4}]$ to classify the states as oscillatory or non-oscillatory. The red dashed line corresponds to the critical threshold $D_{v}^{Crit}$ for Turing instability as obtained from Eq. (14). The region below the critical line corresponds to the parameter values for which oscillatory dynamics is present, whereas in the region above the critical line steady state dynamics persists. For comparatively smaller values of diffusion coefficients $D_{u},D_{v}$, spiral wave pattern is noticed. The corresponding parameter region is shaded with sky blue color. Higher values of prey and predator diffusion strengths (the region shaded in pink) below the red dashed line lead to a pattern where completely synchronized temporal oscillations are obtained. The region shaded in yellow corresponds to the Turing instability region where the system settles down to inhomogeneous steady state dynamics confirming the appearance of various Turing patterns. From the figure it is evident that the analytically derived boundary (marked with red dashed line) partitioning the oscillatory and the non-oscillatory regimes is in perfect accordance with the numerically simulated parameter space.

In the following Fig. 6, we demonstrate the spatial snapshots of the various patterns as mentioned in Fig. 5 and their evolution over time. Left, middle and right columns correspond to the evolution of the prey species exhibiting spiral, synchronized oscillatory and inhomogeneous steady state dynamics, respectively. Snapshots of these patterns in the prey population over a two-dimensional diffusive medium are presented in Figs. 6(a,b,c) at a particular instant of time. The corresponding colorbars quantify the densities of the prey population $u$ at that time instant. To better understand the organization among the species for each pattern we have taken a horizontal cross-section along the central line and plotted them in Figs. 6(d,e,f) with respect to the number of discretized grids in each direction. Figures 6(g,h,i) in the third row exhibit the temporal dynamics of the prey species along the horizontal cross section and finally, Figs. 6(j,k,l) in the bottom row correspond to the spatiotemporal evolution of the patterns among the prey species along the cross-section. Existence of stable spiral pattern is depicted in Fig. 6(a) for diffusion coefficients $D_{u}=0.01$ and $D_{v}=0.001$. The frequency of rotation of the spiral wave decreases and the wavelength of the spiral increases when either one or both the diffusion strengths are increased (figure not shown here). The one dimensional cross-section of the spiral pattern presented in Fig. 6(d) reflects a coherent oscillatory structure among the species in the discrete grids. Temporal dynamics in the form of limit cycle oscillations of these species along the cross section is clearly visible in Fig. 6(g). The observed spiral wave starts to propagate from the core and expands in the outward direction which can be perceived from the spatiotemporal evolution of the prey species along the cross-section depicted in Fig. 6(j). Figure 6(b) reveals the uniform spatial density of the prey species in the two-dimensional medium for diffusion strengths $D_{u}=1.0$ and $D_{v}=0.1$ and their corresponding horizontal cross-section along the center line is presented in Fig. 6(e). The temporal as well as the spatiotemporal evolutions depicted in Figs. 6(h) and 6(k) confirm the existence of completely synchronized limit cycle oscillations with uniform spatial density of all the species. These two figures clearly depict that at any particular instant of time the spatial densities of all the species remain constant. That is why the spatial snapshot in Fig. 6(b) takes only a single value corresponding to the light green color from the colorbar. Emergence of inhomogeneous steady states in the two-dimensional diffusive environment is demonstrated in Fig. 6(c) resulting from the Turing instability at the diffusion coefficient values $D_{u}=0.001$ and $D_{v}=1.0$. Horizontal cross-section of the prey species $u$ along the center line presented in Fig. 6(f) signifies that the species density keeps fluctuating between the upper and the lower branches while maintaining a steady state temporal dynamics. Two distinguishable clusters of prey densities are clearly visible from the temporal dynamics presented in Fig. 6(i). Such kind of temporal pattern where the inhomogeneous steady states are subdivided into more than one distinguishable clusters are known as multicluster oscillation death states mcod . The corresponding spatiotemporal evolution of these multicluster oscillation death states is portrayed in Fig. 6(l).

Besides the above mentioned three distinct categories of spatial patterns, another fascinating spatial organization among the dynamical elements is noticed as the parameter $a$ is shifted away from the Hopf bifurcation boundary. For appropriate choices of prey and predator diffusion strengths $D_{u}$ and $D_{v}$, the core of the spiral motion in the two-dimensional space starts to expand. The dynamical units inside the spiral core exhibit anomalous behavior which differentiates the core region from that of the spiral arm where the constituents follow a smooth motion. Simultaneous occurrence of coherent dynamics in the spiral arm and incoherent dynamics in the spiral core discriminates the spatial pattern from that of the ordinary spiral dynamics. Such coexistence of coherent and incoherent behaviors within a spiral structure develops the captivating spiral chimera pattern. Emergence of spiral chimera patterns in $D_{u}-D_{v}$ parameter space is manifested in Fig. 7 through the parametric region shaded with green color. Other parameter regions shaded with sky blue, pink and yellow colors correspond to similar spatial dynamics as discussed earlier in Fig. 5. Apart from the diffusion strengths associated with the yellow shaded region all other diffusion strengths correspond to oscillatory dynamics of the considered reaction-diffusion system. Two different parameter spaces are depicted in Figs. 7(a) and 7(b), respectively, for two different values of the multiplicative factor $a=4.0$ and $a=6.0$. As mentioned already, the red dashed line delimits the analytically obtained bound $D_{v}^{Crit}$ for Turing instability. Although, contrary to the case of Fig. 5 where the red boundary perfectly matches with the numerically simulated parameter region, a discrepancy is noticed in Fig. 7 between the analytical and numerical bounds separating the oscillatory and non-oscillatory regimes. From the figure it is visible that the analytically derived red boundary fails to determine the critical threshold beyond which oscillatory dynamics vanishes and steady state dynamics (yellow shaded region) begins. Such inconsistency is triggered by the simultaneous occurrence of both the Hopf and the Turing instabilities for the considered parameter values of $a$. In the right hand side of the blue vertical line displayed in Fig. 3, whenever $a$ remains very close to the critical value $a_{0}$, the emergence of Turing instability overrules the Hopf instability, i.e., the oscillatory dynamics is suppressed and the steady state dynamics arises as soon as the diffusion coefficients satisfy the Turing instability condition. That is why for $a=2.0$, once the diffusion coefficients reach the Turing instability condition (Eq. (14)), steady state dynamics replaces the oscillatory dynamics which is reflected perfectly in the parameter space of Fig. 5. However, when $a$ moves away further from $a_{0}$, the appearance of oscillations due to Hopf bifurcation reveals dominating characteristic and as a result, Eq. (14) fails to predict the emergence of steady state dynamics of the system (1). This scenario is demonstrated through the parameter spaces in Fig. 7 as there is appreciable difference between the numerically obtained parameter values (yellow region) and the analytically derived bounds for the appearance of steady state dynamics.

The interplay between the two different instabilities where the oscillations caused by the Hopf bifurcation interact with the inhomogeneous steady states arising due to the Turing bifurcation leads to a fascinatingly complex spiral chimera pattern. We carry out a detailed numerical simulation to analyze the characteristics of such an exotic chimera structure as presented in Fig. 8. Figure 8(a) depicts the typical snapshot of the spiral chimera pattern at a particular time instant $t=5\times 10^{4}$ for $a=4.0$, when the diffusion coefficients are fixed at $D_{u}=0.01$ and $D_{v}=1.0$. For a better understanding of the correlation among the species we take a horizontal cross-section along the center line and plot them in Fig. 8(d). Anomalous species densities around the spiral core surrounded by coherent densities of the prey species is detectable from both the Figs. 8(a) and 8(d). To get further insight about the amplitude of the oscillatory prey densities over the two-dimensional medium, we calculate the finite-time average of the prey densities $u(x,y,t)$ at each grid index $(x,y)$ over $T=10^{3}$ time iterations using the formula $\langle u(x,y,t)\rangle=\frac{1}{T}\int_{0}^{T}u(x,y,t)dt$. We plot the estimated average prey densities $\langle u\rangle$ in the two-dimensional $x-y$ plane in Fig. 8(b) which depicts that apart from the species situated in the spiral core all the remaining prey species oscillate with a nearly identical amplitude, whereas random fluctuations are noticed around the core region. For a clear visualization here also we take the horizontal cross-section along the center line and plot the amplitude differences between two adjacent species densities in Fig. 8(e). Zero values of the amplitude difference $\Delta\langle u\rangle$ suggest coherent amplitude of the prey species densities outside the spiral core while non-zero values of $\Delta\langle u\rangle$ around the core corresponds to incoherent species amplitude. The corresponding spatiotemporal evolution of the species densities along the horizontal cross-section is presented in Fig. 8(c) which reveals that the observed chimera pattern is stationary with respect to time. We further calculate the phases of each of the individual components available in the two-dimensional diffusive medium using the formula $\phi(x,y)=\tan^{-1}\Big{(}\frac{v(x,y)-v_{s}}{u(x,y)-u_{s}}\Big{)}$. Figure 8(f) delineates the phase differences $\Delta\phi$ between two adjacent units situated along the cross-sectioned profile in Fig. 8(d). Again a zero value of $\Delta\phi$ characterizes coherent subpopulation whereas a non-zero value of the phase difference confirms the presence of incoherent dynamics. Thus the observed chimera pattern exhibits coherence-incoherence dynamics in both the phases and the amplitudes of the constituent elements, and hence resembles the structure of an amplitude mediated chimera state. Besides, we also explore the temporal dynamics of each of the individual elements along the cross-sectioned profile. From the cross-sectioned prey densities it is evident that the coherent subpopulations are separated into two disjoint clusters by the incoherent subpopulations. The time series for the coherent subpopulations are presented in Fig. 8(g), where the two distinct clusters follow two distinguishabe temporal patterns marked with blue and black curves. The figure indicates that the two disjoint clusters of coherent subpopulations are in opposite phases with each other. However, the randomness in the temporal dynamics of the incoherent subpopulations is presented by red curves in Fig. 8(h). The corresponding phase space dynamics of the amplitude mediated chimera state along the cross-section is portrayed in Fig. 8(i). Positive correlation among the coherent subpopulations are prominent from the attractors plotted with blue and black curves, while randomness is detectable among the incoherent group through the plotted attractors in red.

Further, we also scrutinize the development of other oscillatory patterns for $a=4.0$ in terms of their average amplitude as illustrated in Fig. 9. The upper panel depicts the typical snapshots of distinct spatial patterns that are oscillating in time for different combinations of the diffusion strengths at a particular time instant $t=5\times 10^{4}$. The finite-time averages of each of these individual patterns over $T=10^{3}$ time iterations are presented in the middle row, while the amplitude differences between the adjacent nodes along the horizontal cross-sections through the center lines are presented in the bottom row. Snapshot of an ordinary spiral is portrayed in Fig. 9(a) for lower values of both the diffusion coefficients $D_{u}=D_{v}=0.01$. Figure 9(d) shows the corresponding time average $\langle u\rangle$ of each individual prey species in the two-dimensional space. The figure reveals that except at the center where the average amplitude is low (around $2.3$) compared to the remaining space that is occupied with average amplitude of species ranging between $2.5$ to $2.6$. The amplitude difference $\Delta\langle u\rangle$ plotted in Fig. 9(g) along the horizontal cross-section through $y=124$ takes approximately zero value for all the spatial positions except for very few nodes at the center. The frequency of rotation of the spiral wave decreases as one or both the diffusion strengths are increased and an exemplary snapshot of the spatial pattern is presented in Fig. 9(b) for diffusion strengths $D_{u}=0.01$ and $D_{v}=0.1$. The corresponding finite-time average $\langle u\rangle$ represented in Fig. 9(e) again confirms that the species densities oscillate with nearly identical amplitude throughout the two-dimensional space except at the center. However, the size of the core with drifted amplitude starts to increase as the diffusion strengths increase which is apparent from Fig. 9(h). Finally, Fig. 9(c) in the right panel corresponds to uniform spatial densities of the prey species throughout the two-dimensional space for $D_{u}=0.1$ and $D_{v}=1.0$. The identical amplitude of oscillating species densities throughout the space is easily recognizable from the Figs. 9(f) and 9(i).

Next, we also demonstrate the spatial arrangement as well as the spatiotemporal organization of the prey densities when they are in the steady state regime. The emergence of Turing instability for suitable combinations of prey and predator diffusion coefficients drives the system to exhibit inhomogeneous steady state dynamics. The transition from amplitude mediated spiral chimeras to inhomogeneous steady states results in two different types of oscillation death states. One is the chimera death state, where a coexistence of coherent and incoherent oscillation death states is observed just like the oscillatory chimera states, and the other is the incoherent oscillation death state, where no coherent behavior is detected among the constituents. Two distinct snapshots of the spatial organization among the prey species in oscillation death state are presented in Figs. 10(a) and 10(b) for diffusion strengths $D_{u}=0.0018,D_{v}=1.0$ and $D_{u}=0.0056,D_{v}=1.0$, respectively. In the middle panel the corresponding snapshots along the horizontal cross-sections through the line $y=125$ are presented. Coexistence of coherent and incoherent oscillation death states is prominent from Fig. 10(c) which characterizes the state as chimera death state. On the other hand, Fig. 10(d) clearly demonstrates the presence of incoherent oscillation death states which keeps on fluctuating randomly between the upper and the lower branches of the inhomogeneous steady states. The stationary behavior of such states is further confirmed from the spatiotemporal dynamics presented in Figs. 10(e) and 10(f), respectively.

In order to characterize the occurrence of all such diverse patterns, we introduce a measure to estimate the strength of incoherent populations in a particular pattern arising in the two-dimensional space. On the basis of the computed finite-time average of the species densities for each pattern, we calculate the strength of incoherent subpopulation for each of those dynamical patterns using the following relation

generalizing the notion of strength of incoherence introduced by Gopal et al. chimera_nonlocal3 to characterize chimera states in one dimensional arrays. Here $\bar{u}(x,y)=\langle u(x,y)\rangle-\langle u_{c}\rangle$ measures the deviation of each of the species densities situated at position $(x,y)$ from that of the coherent subpopulations. $\langle u_{c}\rangle$ represents the average amplitude of the coherent subpopulations throughout the two-dimensional space which is kept fixed at $\langle u_{c}\rangle=2.55$. $\bar{u}=0$ corresponds to the coherent subpopulation whereas the nonzero values of $\bar{u}$ signify the species belonging to the incoherent group. $\delta$ is a predefined threshold value and $\Theta(\cdot)$ denotes the Heaviside step function.

The measure for the estimation of strength of incoherent subpopulations $S$ lies in the range $[0,1]$. Zero value of $S$ is attained when the system exhibits completely synchronized temporally oscillating uniform spatial densities of the species, while $S=1$ stands for the oscillation death states or the inhomogeneous steady states. Various spiral dynamics are represented by the non-zero and non-unit values of $S$, i.e., $0<S<1$. However, for ordinary spirals due to the presence of very little incoherence around the core, $S$ takes values close to zero. Existence of amplitude mediated spiral chimera states is determined by sufficiently large values of $S$ lying between $0$ and $1$. In Fig. 11 the transitions among different dynamical states are illustrated by plotting the measure $S$ with respect to the diffusion coefficients $D_{u},D_{v}$. Three distinct collective dynamics, namely oscillation death (OD), spiral chimera (SC) and completely synchronized oscillation (CS), are observed when the strength of incoherent subpopulations are plotted in Fig. 11(a) with respect to varying prey diffusion rate $D_{u}$ for fixed value of the predator diffusion rate $D_{v}=0.56$. Additionally, in Fig. 11(b) we plot the measure $S$ for varying predator diffusion strength $D_{v}$ at fixed value of prey diffusion strength $D_{u}=0.01$. Here also three distinct dynamical states are detected, namely spiral wave (SW), spiral chimera (SC) and oscillation death (OD) states. From the figure it is evident that the amplitude mediated spiral chimera states appear during the transition from oscillatory motion to steady state dynamics.

Throughout the article, all the numerical simulations are carried out using the fixed grid size $N^{2}=249^{2}$, spatial step $h=0.25$ and integration time step $dt=0.025$. Now, we check the effect of varying $N$ and $h$ on the observed patterns of the considered system. For this, we fix the diffusion coefficients at $D_{u}=0.01$ and $D_{v}=1.0$ for which spiral chimera pattern was observed in Fig. 8 with $N=249$. Now, if the value of $N$ is reduced gradually, the dynamics changes from spiral chimera to completely synchronized oscillation states (figure not shown), whereas increasing $N$ retains the dynamics unchanged as obtained in Fig. 8(a). It should be noted that varying only $N$ alters the spatial distance $N\times h$. Thus, lowering $N$ reduces the spatial distance which may not be sufficient for the emergence of the spiral pattern which is why the dynamics gets changed for smaller values of $N$. Figure 12(a) displays the snapshot of the spatial evolution of the pattern obtained for $N=401$ at a particular time instant $t=5\times 10^{4}$. Obviously, the pattern is same as depicted in Fig. 8(a) since the spatial length $N\times h$ is sufficiently large here. On the other hand, we have also verified that similar spatial patterns can be observed by varying the spatial step size $h$, but the time step $dt$, time iteration and $N$ should also be changed accordingly. For example, we choose $h=0.15,dt=0.015,N=415$ and time iteration $=3.5\times 10^{6}$ such that the spatial distance $N\times h$ and the time $t$ (iteration$\times dt$) at which the snapshot is recorded remains almost unchanged. The snapshot of the spatial pattern is depicted in Fig. 12(b). Therefore, other choices of $N,h,dt$ may also give rise to the similar dynamics as long as the spatial distance $N\times h$ and the time iteration $t/dt$ remains sufficient for the emergence of the respective patterns.