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Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application.

Karnatak R - PLoS ONE (2015)

Bottom Line: The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed.Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios.Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

View Article: PubMed Central - PubMed

Affiliation: Nonlinear Dynamics and Time Series Analysis Research Group, Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.

ABSTRACT
Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

No MeSH data available.


Different dynamical regimes for H II and H III systems.Regimes are marked as A, B and C in the ε, k plane. A is the regime of periodic dynamics, in B stationary solutions of the coupled system are stable and C is the regime of stable origin. The boundaries between A→B and B→C are the loci of the reverse Hopf bifurcation H and the second transcritical bifurcation T2 respectively (as in Fig 4: top row).
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pone.0142238.g005: Different dynamical regimes for H II and H III systems.Regimes are marked as A, B and C in the ε, k plane. A is the regime of periodic dynamics, in B stationary solutions of the coupled system are stable and C is the regime of stable origin. The boundaries between A→B and B→C are the loci of the reverse Hopf bifurcation H and the second transcritical bifurcation T2 respectively (as in Fig 4: top row).

Mentions: Now considering applications, as already mentioned, origin corresponds to an equilibrium for which the predators and the preys vanish. Persistence of populations for a proper ecosystem function is very imperative and has been studied extensively from several perspectives, contributing towards a better understanding of the processes leading to species extinction [48–52]. Knowledge regarding these processes can help in devising procedures which can contribute towards better species conservation efforts. For the simple models considered in the previous analysis, it is clear that either by coupling the system appropriately or by using specific parameter values for k and ε, we can avoid stabilizing the origin. For instance, considering the prey augmented case, for low ε, the systems exhibit periodic oscillations. On increasing the coupling strength, stationary solutions of the augmented system (x* > 0, y* > 0, u* < 0), satisfyingrx*(1-x*/K)-f(x*)y*+ε1u*=0,(ρf(x*)-γ)y*=0,-ku*-εx*=0,(23)get stabilized through a reverse Hopf bifurcation (marked as H in Fig 4 (top row)). For these stationary solutions, the value of x* stays constant while y* and u* = −εx*/k show a variation for a range of ε values (plateau between H and T1 in Fig 4 (top row)). It is also important to note that some initial conditions in this regime can lead to trajectories escaping to infinity. This branch of solutions undergoes a transcritical bifurcation (marked T1 in Fig 4 (top row)) where it exchanges stability with another branch of solutions with u* → 0 for increasing ε. At , u* = 0 and the predator—prey system effectively decouples from augmentation which is accompanied by another transcritical bifurcation (T2 in Fig 4 (top row)) between the continuing branch of stationary solutions (x* > 0, y* > 0, u* → 0) and the origin. In ε > ε* regime, origin is the only dynamical attractor. Fig 5 shows the parameter scans for H II and H III systems highlighting these different dynamical regimes. In region A these systems exhibit periodic behavior and the boundary between A and B is the locus of the Hopf bifurcation in the ε, k plane, which leads to the stabilization of stationary solutions (x* > 0, y* > 0, u* < 0). B corresponds to the regime where stable stationary solutions (x* > 0, y* > 0, u* < 0) and (x* > 0, y* > 0, u* → 0) are observed and the boundary between B and C is the locus of the second transcritical bifurcation T2 which leads to the stabilization of the origin. Therefore by using appropriate values of ε and k, we can keep the system in either a periodic state, or a stationary state with non vanishing populations and can expect this control to work in experiments and be robust with respect to demographic noise; Ref. [26] experimentally stabilized a stationary solution in an electronic Lorenz system at permitted noise level. Furthermore, in the other instances of augmented predators, or augmented predators and preys we already observe a complete lack of origin stabilization. Therefore, we can employ these schemes as well to avoid stabilizing the origin but one needs to be careful since these cases can lead to other complications as discussed. Another useful application for these observations could be in cases where maximization of prey yield is required. Augmenting the predator populations is seen to stabilize the equilibrium where the prey populations exist at their carrying capacity and the predators vanish. This can find applications in fisheries [53, 54], algae fuel generation [55, 56]; where maximal sustainable yields are crucial, and also in biomedical research, for e.g. in HIV-1 infection models [57] where a portion of human immune system i.e. activated CD4+ T cells are the primary target of the HIV-1 infection [58, 59] which can be modeled via predator—prey dynamics.


Linear Augmentation for Stabilizing Stationary Solutions: Potential Pitfalls and Their Application.

Karnatak R - PLoS ONE (2015)

Different dynamical regimes for H II and H III systems.Regimes are marked as A, B and C in the ε, k plane. A is the regime of periodic dynamics, in B stationary solutions of the coupled system are stable and C is the regime of stable origin. The boundaries between A→B and B→C are the loci of the reverse Hopf bifurcation H and the second transcritical bifurcation T2 respectively (as in Fig 4: top row).
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4636295&req=5

pone.0142238.g005: Different dynamical regimes for H II and H III systems.Regimes are marked as A, B and C in the ε, k plane. A is the regime of periodic dynamics, in B stationary solutions of the coupled system are stable and C is the regime of stable origin. The boundaries between A→B and B→C are the loci of the reverse Hopf bifurcation H and the second transcritical bifurcation T2 respectively (as in Fig 4: top row).
Mentions: Now considering applications, as already mentioned, origin corresponds to an equilibrium for which the predators and the preys vanish. Persistence of populations for a proper ecosystem function is very imperative and has been studied extensively from several perspectives, contributing towards a better understanding of the processes leading to species extinction [48–52]. Knowledge regarding these processes can help in devising procedures which can contribute towards better species conservation efforts. For the simple models considered in the previous analysis, it is clear that either by coupling the system appropriately or by using specific parameter values for k and ε, we can avoid stabilizing the origin. For instance, considering the prey augmented case, for low ε, the systems exhibit periodic oscillations. On increasing the coupling strength, stationary solutions of the augmented system (x* > 0, y* > 0, u* < 0), satisfyingrx*(1-x*/K)-f(x*)y*+ε1u*=0,(ρf(x*)-γ)y*=0,-ku*-εx*=0,(23)get stabilized through a reverse Hopf bifurcation (marked as H in Fig 4 (top row)). For these stationary solutions, the value of x* stays constant while y* and u* = −εx*/k show a variation for a range of ε values (plateau between H and T1 in Fig 4 (top row)). It is also important to note that some initial conditions in this regime can lead to trajectories escaping to infinity. This branch of solutions undergoes a transcritical bifurcation (marked T1 in Fig 4 (top row)) where it exchanges stability with another branch of solutions with u* → 0 for increasing ε. At , u* = 0 and the predator—prey system effectively decouples from augmentation which is accompanied by another transcritical bifurcation (T2 in Fig 4 (top row)) between the continuing branch of stationary solutions (x* > 0, y* > 0, u* → 0) and the origin. In ε > ε* regime, origin is the only dynamical attractor. Fig 5 shows the parameter scans for H II and H III systems highlighting these different dynamical regimes. In region A these systems exhibit periodic behavior and the boundary between A and B is the locus of the Hopf bifurcation in the ε, k plane, which leads to the stabilization of stationary solutions (x* > 0, y* > 0, u* < 0). B corresponds to the regime where stable stationary solutions (x* > 0, y* > 0, u* < 0) and (x* > 0, y* > 0, u* → 0) are observed and the boundary between B and C is the locus of the second transcritical bifurcation T2 which leads to the stabilization of the origin. Therefore by using appropriate values of ε and k, we can keep the system in either a periodic state, or a stationary state with non vanishing populations and can expect this control to work in experiments and be robust with respect to demographic noise; Ref. [26] experimentally stabilized a stationary solution in an electronic Lorenz system at permitted noise level. Furthermore, in the other instances of augmented predators, or augmented predators and preys we already observe a complete lack of origin stabilization. Therefore, we can employ these schemes as well to avoid stabilizing the origin but one needs to be careful since these cases can lead to other complications as discussed. Another useful application for these observations could be in cases where maximization of prey yield is required. Augmenting the predator populations is seen to stabilize the equilibrium where the prey populations exist at their carrying capacity and the predators vanish. This can find applications in fisheries [53, 54], algae fuel generation [55, 56]; where maximal sustainable yields are crucial, and also in biomedical research, for e.g. in HIV-1 infection models [57] where a portion of human immune system i.e. activated CD4+ T cells are the primary target of the HIV-1 infection [58, 59] which can be modeled via predator—prey dynamics.

Bottom Line: The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed.Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios.Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

View Article: PubMed Central - PubMed

Affiliation: Nonlinear Dynamics and Time Series Analysis Research Group, Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.

ABSTRACT
Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the main highlight of this scheme but questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Examples from conservative as well as dissipative systems are presented in this regard and important applications in dissipative predator-prey systems are discussed, which include preventative measures to avoid potentially catastrophic dynamical transitions in these systems.

No MeSH data available.