Bounds on the dynamics of sink populations with noisy immigration

Eager, Eric Alan; Guiver, Chris; Hodgson, David J.; Rebarber, Richard; Stott, Iain; Townley, Stuart

doi:10.1016/j.tpb.2013.12.004

Cited by 6 publications

(6 citation statements)

References 44 publications

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“…We refer the reader to Eager et al. ( 2014 ) and the references therein for more information on the impacts of nonnegative disturbances on populations modelled by matrix PPMs.…”

Section: Multi-input Multi-output Low-gain Pi Control For Positive Smentioning

confidence: 99%

Robust set-point regulation for ecological models with multiple management goals

et al. 2015

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Population managers will often have to deal with problems of meeting multiple goals, for example, keeping at specific levels both the total population and population abundances in given stage-classes of a stratified population. In control engineering, such set-point regulation problems are commonly tackled using multi-input, multi-output proportional and integral (PI) feedback controllers. Building on our recent results for population management with single goals, we develop a PI control approach in a context of multi-objective population management. We show that robust set-point regulation is achieved by using a modified PI controller with saturation and anti-windup elements, both described in the paper, and illustrate the theory with examples. Our results apply more generally to linear control systems with positive state variables, including a class of infinite-dimensional systems, and thus have broader appeal.

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“…We refer the reader to Eager et al. ( 2014 ) and the references therein for more information on the impacts of nonnegative disturbances on populations modelled by matrix PPMs.…”

Section: Multi-input Multi-output Low-gain Pi Control For Positive Smentioning

confidence: 99%

Robust set-point regulation for ecological models with multiple management goals

et al. 2015

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“…The current work continues a line of enquiry of the present authors, including Eager et al. (2014) and Franco et al. (2017), to develop a theoretical framework for the analysis of the effects of forcing in ecological modelling, analysis, and control.…”

Section: Discussionmentioning

confidence: 68%

“…Boundedness is a necessary property of a sensible biological model and, moreover, is a key concept when seeking to understand the potential effects of an invasive species or mutant in a novel environment, see, for example, Eager et al. (2014). The notion of persistence relates to the survival of a population (or survival of certain stage-classes) and is relevant, for example, in the context of providing lower bounds for predicted yield in agriculture and horticulture.…”

Section: Introductionmentioning

confidence: 99%

Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

et al. 2019

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Boundedness, persistence and stability properties are considered for a class of nonlinear, possibly infinite-dimensional, forced difference equations which arise in a number of ecological and biological contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes), disturbances induced by seasonal or environmental variation, or migration. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We present sufficient conditions for the non-zero equilibrium to be stable in a sense which is strongly inspired by the input-to-state stability concept well-known in mathematical control theory. In particular, our stability concept incorporates the impact of potentially persistent forcing. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). The theory is applied to a number of examples from population dynamics.

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“…While the mean time to extinction and quasi-stationary state are central parameters for predicting the fate of an established (meta)population, the influence of immigration, in particular the process of colonization, also depends on the probability of reaching the established state (Grimm & Wissel 2004), called the establishment curve if considered a function of initial population size (Drake & Lodge 2006). Since the pioneering works of island biogeography (MacArthur & Wilson 1963, Brown & Kodric-Brown 1977, the influence of immigration on local dynamics has been studied from several points of view: immigration may stabilize chaotic behavior (Ruxton 1995, Stone & Hart 1999, it may sustain populations in sink patches (Eager et al 2014), and influence demographic (Matis & Kiffe 1999 and genetic (Robert et al 2003) population dynamics and consequently metapopulation dynamics (Lande et al 1998). Intuitively, if the immigration rate is much smaller (or conversely, much larger) than the extinction rate, the population is likely to be extinct (or conversely, extant) for the majority of time.…”

Section: Ilkka Hanski: the Legacy Of A Multifaceted Ecologistmentioning

confidence: 99%

The Interplay between Immigration and Local Population Dynamics in Metapopulations

Ovaskainen

2017

Annales Zoologici Fennici

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Stochastic models of closed populations predict eventual extinction with certainty. Consequently, their behavior is often characterized by the quasi-stationary state, i.e. the long-term distribution of population sizes conditional on non-extinction. In contrast, models which allow for immigration exhibit a regular stationary state. At the limit of a low immigration rate, a population is expected to alternate between three states: the quasi-stationary state of a closed population, the extinction state, and the transient phase during which a newly arrived immigrant either establishes a new population or fails to do so. We develop this argument into a simple and intuitive framework that can be used to assess the effect of immigration in a general class of population models. We exemplify the framework for models in which immigrants arrive either singly or in groups, for models with an Allee effect, for models with environmental stochasticity, and for models leading to metapopulation dynamics.

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Bounds on the dynamics of sink populations with noisy immigration

Cited by 6 publications

References 44 publications

Robust set-point regulation for ecological models with multiple management goals

Robust set-point regulation for ecological models with multiple management goals

Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

The Interplay between Immigration and Local Population Dynamics in Metapopulations

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