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Applied Mathematics

Dynamics of an ecological model living on the edge of chaos

Profile image of Ranjit Kumar UpadhyayRanjit Kumar Upadhyay

2009, Applied Mathematics and Computation

https://doi.org/10.1016/J.AMC.2009.01.006
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Abstract

We present a new ecological model, which displays ''edge of chaos" (EoC) in parameter space. This suggests that ecological systems are not chaotic, instead, their dynamics can be characterized as short-term recurrent chaos. The system's dynamics is unpredictable and admits bursts of short-term predictability. We also provide results, which suggest that fully developed chaos will rarely be observed in natural systems.

Figures (11)
Fig. 1. Time-series displaying oscillatory dynamics in modified RM model (1a) and (1b) for the parameter values a, = 2, b, = 0.06, w= 0.85, « = 0.45, f = 0.2 y = 0.6, a2 = 1.05 and w, = 1.65. The best way to design some mathematical models of ecological systems is to couple the two subsystems — modified RM model and modified HT model - using one of the fundamental interactions: predation, competition, interference and mutualism. This design facilitates the selection of parameter values for simulation experiments. When the values of the An oscillatory predator-prey dynamics exhibited by the model system (2a) and (2b) ata typical set of parameter values is presented in Fig. 2.
Fig. 1. Time-series displaying oscillatory dynamics in modified RM model (1a) and (1b) for the parameter values a, = 2, b, = 0.06, w= 0.85, « = 0.45, f = 0.2 y = 0.6, a2 = 1.05 and w, = 1.65. The best way to design some mathematical models of ecological systems is to couple the two subsystems — modified RM model and modified HT model - using one of the fundamental interactions: predation, competition, interference and mutualism. This design facilitates the selection of parameter values for simulation experiments. When the values of the An oscillatory predator-prey dynamics exhibited by the model system (2a) and (2b) ata typical set of parameter values is presented in Fig. 2.
Fig. 2. Time-series displaying oscillatory predator-prey dynamics exhibited by the modified HT model (2a) and (2b) for the parameter values A = 2, K = 100, w3 = 2.1, % =0.5, B, = 0.25, y; = 0.65, c= 0.55 and wy = 1.0.
Fig. 2. Time-series displaying oscillatory predator-prey dynamics exhibited by the modified HT model (2a) and (2b) for the parameter values A = 2, K = 100, w3 = 2.1, % =0.5, B, = 0.25, y; = 0.65, c= 0.55 and wy = 1.0.
parameters are chosen in such a way that both modified RM and modified HT model systems display regular persistent peri- odic behavior, these systems behave as oscillators. Let us call modified RM oscillator as LCO (1) and modified HT oscillator as LCO (2). When suitably coupled, these systems force each other to generate chaos. An ecologically sound coupling of these systems gives us the following model system: All the parameters are positive and have similar meaning as used in the subsystems. The last term in Eq. (3b) represents the functional response of the predator U, which is a generalist predator. It switches its prey whenever its favorite food option Z is in short supply. The last term in Eq. (3d) describes how loss in species U depends on per capita availability of its preys (Y and Z). wz serves as the coupling parameter.
parameters are chosen in such a way that both modified RM and modified HT model systems display regular persistent peri- odic behavior, these systems behave as oscillators. Let us call modified RM oscillator as LCO (1) and modified HT oscillator as LCO (2). When suitably coupled, these systems force each other to generate chaos. An ecologically sound coupling of these systems gives us the following model system: All the parameters are positive and have similar meaning as used in the subsystems. The last term in Eq. (3b) represents the functional response of the predator U, which is a generalist predator. It switches its prey whenever its favorite food option Z is in short supply. The last term in Eq. (3d) describes how loss in species U depends on per capita availability of its preys (Y and Z). wz serves as the coupling parameter.
The following diagram helps us to understand the model system in terms of its subsystems. The dynamics of specialist predator and its prey is described by modified RM system. The dynamics of generalist predator and its pseudo-prey is gov- erned by modified HT system. The coupling of these two systems is achieved by feeding of the generalist predator on its most favorite food (the specialist predator). The selection of parameter values for simulation experiments are carried out in such a way that both the subsystems of nodel system described by Eqs. (4a, 4b, and 5) (subsystem A is given by Eqs. (1a and 1b) and subsystem B is described by =qs. (2a and 2b)) evolve on stable limit cycles. Model parameters are selected in accordance with a method given by Upad- iyay et al. [13,16]. We study the asymptotic dynamics of the model system in two-dimensional parameter space. We have 9erformed two-dimensional (2D) parameter scans to identify the parameter regimes in which different dynamics (stable fo- cus, Stable limit cycle and chaos) exist. The parameters chosen for the 2D scans are the parameters which control the dynam- cs of the model system. The basis for performing the 2D scans is the belief that changes in physical conditions may bring corresponding changes in at least two parameters at a time. Two parameters are varied on both sides of their limit cycle val- 1es. The ranges for variation of these control parameters in simulation experiments are decided on the available biological nformation. All the other parameters are given biologically realistic values taken from the compilation of Jorgensen et al. 19]. The model system was simulated for different initial conditions at all sets of parameter values, where chaos was ybserved.
The following diagram helps us to understand the model system in terms of its subsystems. The dynamics of specialist predator and its prey is described by modified RM system. The dynamics of generalist predator and its pseudo-prey is gov- erned by modified HT system. The coupling of these two systems is achieved by feeding of the generalist predator on its most favorite food (the specialist predator). The selection of parameter values for simulation experiments are carried out in such a way that both the subsystems of nodel system described by Eqs. (4a, 4b, and 5) (subsystem A is given by Eqs. (1a and 1b) and subsystem B is described by =qs. (2a and 2b)) evolve on stable limit cycles. Model parameters are selected in accordance with a method given by Upad- iyay et al. [13,16]. We study the asymptotic dynamics of the model system in two-dimensional parameter space. We have 9erformed two-dimensional (2D) parameter scans to identify the parameter regimes in which different dynamics (stable fo- cus, Stable limit cycle and chaos) exist. The parameters chosen for the 2D scans are the parameters which control the dynam- cs of the model system. The basis for performing the 2D scans is the belief that changes in physical conditions may bring corresponding changes in at least two parameters at a time. Two parameters are varied on both sides of their limit cycle val- 1es. The ranges for variation of these control parameters in simulation experiments are decided on the available biological nformation. All the other parameters are given biologically realistic values taken from the compilation of Jorgensen et al. 19]. The model system was simulated for different initial conditions at all sets of parameter values, where chaos was ybserved.
Simulation experiments of the model system given by Eqs. (4a, 4b, and 5) with parameter values given in Eq. (6). Table 1
Simulation experiments of the model system given by Eqs. (4a, 4b, and 5) with parameter values given in Eq. (6). Table 1
Simulation experiments of the model system given by Eqs. (4a, 4b, and 5) with parameter values given in Eq. (6). Table 2 There may exist other sets of parameter values which satisfy the above criteria. Chaos does not exist in this model system at all. The domain in which we performed the two-dimensional scans are the following:
Simulation experiments of the model system given by Eqs. (4a, 4b, and 5) with parameter values given in Eq. (6). Table 2 There may exist other sets of parameter values which satisfy the above criteria. Chaos does not exist in this model system at all. The domain in which we performed the two-dimensional scans are the following:
Fig. 3. Distribution of points where chaos was observed in the model system on the parameter spaces. (a) For (aj, /): [0.5 < a; < 3.0, 0.01 < 8 < 0.3] wher a, in step size 0.25 and f# in step size 0.01. (b) For (a,c): [0.5 < a; < 2.5, 0.001 < c < 0.0.036] where a, in step size 0.25 and c in step size 0.001. When the 2D parameter scans are performed using four particular groups of parameters, the values of remaining param- eters are given in (6). The parameters a, and az are varied with step size 0.25 and rest of the parameters in step size 0.01. Fig. 3a and b displays points where deterministic chaos was observed. The abscissa is spanned by the specific rate of growth
Fig. 3. Distribution of points where chaos was observed in the model system on the parameter spaces. (a) For (aj, /): [0.5 < a; < 3.0, 0.01 < 8 < 0.3] wher a, in step size 0.25 and f# in step size 0.01. (b) For (a,c): [0.5 < a; < 2.5, 0.001 < c < 0.0.036] where a, in step size 0.25 and c in step size 0.001. When the 2D parameter scans are performed using four particular groups of parameters, the values of remaining param- eters are given in (6). The parameters a, and az are varied with step size 0.25 and rest of the parameters in step size 0.01. Fig. 3a and b displays points where deterministic chaos was observed. The abscissa is spanned by the specific rate of growth
Fig. 4. Distribution of points where chaos was observed in model system 1 on the parameter spaces. (a) For (dz, 8): [0.01 < az < 2,0.01 < 6 < 0.25] where ap in step size 0.25 and f in step size 0.01. (b) For (az, «): [0.01 < az < 1.5, 0.01 < ~ < 2.5] where ap in step size 0.25 and «@ in step size 0.05.
Fig. 4. Distribution of points where chaos was observed in model system 1 on the parameter spaces. (a) For (dz, 8): [0.01 < az < 2,0.01 < 6 < 0.25] where ap in step size 0.25 and f in step size 0.01. (b) For (az, «): [0.01 < az < 1.5, 0.01 < ~ < 2.5] where ap in step size 0.25 and «@ in step size 0.05.
Fig. 5. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 1.4 < a, < 2.8 with other parameters as given in Eq. (6).
Fig. 5. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 1.4 < a, < 2.8 with other parameters as given in Eq. (6).
Fig. 6. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 0.025 < c < 0.033 with other parameters as given in Eq. (6). f the prey population due to self-reproduction in the domain 0.5 < a; < 3.0 and the ordinates are spanned by parameter: 1easuring the intensity of interference between individuals of the specialist predator and the growth rate of the generalis redator in the domain 0.01 < £ < 2.0 and 0.001 < c< 0.5, respectively. The model system does not display chaotic dynam: ‘s for small value of prey’s per capita self-reproduction rate (i.e., a; < 1.5); however chaotic dynamics may be observed ir ne range 1.5 < a, < 2.5. Fig. 4a and b shows the points where chaos was detected in the parameter spaces (a>,f) and (a2,«) spectively, which aimed at investigating how changes in the mortality rates of the specialist predator alter the eventua ynamics displaced by the model system. From the tables and 2D scan results, we found that as we raised the control param: ters (d4,C,d2,f,a%) the dynamics of the model system went quickly frozen at a fixed point to periodic to eventually chaoti: ynamics. It is important to note that these transitions occurred in a wide variety of the parameter values which could divide ne space between ordered (stable focus and stable limit cycle) and chaotic regimes. The points shown all these figures rep. 2sent edge of chaos phenomena as chaos exists at discrete isolated points in these parameter spaces. From the 2D scan re. ults, it was observed that the chaos was found in the ranges 1.4 < a; < 2.8, 0.025 <c < 0.035 and 0.15 < # < 0.25. Fron able 1, it was found that in the parameter space (a1,c) deterministic chaos was observed at few discrete points. For example haos exists for (a;,c) =(1.5,0.03), (1.5,0.035), (1.75,0.035), (2,0.03) and (2.25,0.03). It was also observed that always stable mit cycle transitions into chaos where complexity is at its maximum. After that no specific dynamics was exhibited by the 10del system and it has been reported as integration error. The integration error was found in the range 0.04 < c < 0.5. Fo! ther parameter spaces, the dynamics belonging to left, right, above and below the chaotic points was found to be the stable
Fig. 6. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 0.025 < c < 0.033 with other parameters as given in Eq. (6). f the prey population due to self-reproduction in the domain 0.5 < a; < 3.0 and the ordinates are spanned by parameter: 1easuring the intensity of interference between individuals of the specialist predator and the growth rate of the generalis redator in the domain 0.01 < £ < 2.0 and 0.001 < c< 0.5, respectively. The model system does not display chaotic dynam: ‘s for small value of prey’s per capita self-reproduction rate (i.e., a; < 1.5); however chaotic dynamics may be observed ir ne range 1.5 < a, < 2.5. Fig. 4a and b shows the points where chaos was detected in the parameter spaces (a>,f) and (a2,«) spectively, which aimed at investigating how changes in the mortality rates of the specialist predator alter the eventua ynamics displaced by the model system. From the tables and 2D scan results, we found that as we raised the control param: ters (d4,C,d2,f,a%) the dynamics of the model system went quickly frozen at a fixed point to periodic to eventually chaoti: ynamics. It is important to note that these transitions occurred in a wide variety of the parameter values which could divide ne space between ordered (stable focus and stable limit cycle) and chaotic regimes. The points shown all these figures rep. 2sent edge of chaos phenomena as chaos exists at discrete isolated points in these parameter spaces. From the 2D scan re. ults, it was observed that the chaos was found in the ranges 1.4 < a; < 2.8, 0.025 <c < 0.035 and 0.15 < # < 0.25. Fron able 1, it was found that in the parameter space (a1,c) deterministic chaos was observed at few discrete points. For example haos exists for (a;,c) =(1.5,0.03), (1.5,0.035), (1.75,0.035), (2,0.03) and (2.25,0.03). It was also observed that always stable mit cycle transitions into chaos where complexity is at its maximum. After that no specific dynamics was exhibited by the 10del system and it has been reported as integration error. The integration error was found in the range 0.04 < c < 0.5. Fo! ther parameter spaces, the dynamics belonging to left, right, above and below the chaotic points was found to be the stable
Fig. 7. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 0.15 < 8 < 0.25 with other parameters as given in Eq. (6).
Fig. 7. Bifurcation diagram of the model system given by Eqs. (4a, 4b, and 5) for 0.15 < 8 < 0.25 with other parameters as given in Eq. (6).

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