Harvesting a logistic population in a slowly varying environment

Idlango, Majda A.; Shepherd, John J.; Nguyen, Le Ha Vy; Gear, John Anthony

doi:10.1016/j.aml.2011.07.015

Cited by 13 publications

(18 citation statements)

References 6 publications

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Section: Introductionmentioning

confidence: 80%

“…Note that when any of R, K or M is constant, its dimensionless counterpart is unity. Thus, if R.T/ is constant, r. t/ Á 1 in the problem (5). Similarly for K.T/ or M.T/.…”

Section: Introductionmentioning

confidence: 94%

“…While t and εt can be viewed as two possible time scales for the problem , as noted in the Introduction, experience with similar problems leads us to a more general approach. Thus, we define the normal time, t 0 and slow time, t 1 by

t_{0} MathClass-rel= \frac{1}{ε} h MathClass-open(t_{1} MathClass-close) and t_{1} MathClass-rel= εt MathClass-punc,

where h ( t 1 ) is a positive valued function on all t 1 > 0 to be determined, and which satisfies the condition h ′ ( t 1 ) > 0 on all t 1 > 0, to ensure one‐to‐one correspondence between t 0 and t ; whereas h (0) = 0.…”

Section: The Multiscale Equationmentioning

confidence: 99%

“…If we are to adopt the approach used in similar problems , we would substitute for

{\overset{MathClass-op̃}{p}}_{0}

in , the equation for

{\overset{MathClass-op̃}{p}}_{1}

, and obtain a particular solution involving the functions A ( t 1 ) and h ′ ( t 1 ) , that, typically would show a convergence rate of

t_{0} e^{- γ (t_{1}) t_{0}}

(or worse). A ( t 1 ) and h ′ ( t 1 ) are then chosen to remove this nonconforming rate, and in this way, we would obtain expressions for these functions and preserve the convergence rate.…”

Section: Perturbation Analysismentioning

confidence: 99%

“…In this case, we can apply a multiscaled perturbation method based on these time scales to give analytic approximations to the solutions of this problem. In this, our analysis will parallel that of [5] In the next section, we will investigate the basic dimensionless Allee model when the parameters are positive constants. In this case, (5) is autonomous, and we obtain an implicit representation for the exact solution for an arbitrary p 0 , as well as expansions for the limiting states of this solution as t !…”

Section: Introductionmentioning

confidence: 99%

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Multiscaling analysis of a slowly varying single species population model displaying an Allee effect

Idlango

Shepherd

Gear

2013

Math Methods in App Sciences

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We consider the growth of a single species population modelled by a logistic equation modified to accommodate an Allee effect, in which the model parameters are slowly varying functions of time. We apply a multitiming technique to construct general approximate expressions for the evolving population in the case where the population survives to a (slowly varying) finite positive limiting state, and that where the population declines to extinction. We show that these expressions give excellent agreement with the results of numerical calculations for particular instances of the changing model parameters. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 80%

“…Note that when any of R, K or M is constant, its dimensionless counterpart is unity. Thus, if R.T/ is constant, r. t/ Á 1 in the problem (5). Similarly for K.T/ or M.T/.…”

Section: Introductionmentioning

confidence: 94%

t_{0} MathClass-rel= \frac{1}{ε} h MathClass-open(t_{1} MathClass-close) and t_{1} MathClass-rel= εt MathClass-punc,

Section: The Multiscale Equationmentioning

confidence: 99%

“…If we are to adopt the approach used in similar problems , we would substitute for

{\overset{MathClass-op̃}{p}}_{0}

in , the equation for

{\overset{MathClass-op̃}{p}}_{1}

, and obtain a particular solution involving the functions A ( t 1 ) and h ′ ( t 1 ) , that, typically would show a convergence rate of

t_{0} e^{- γ (t_{1}) t_{0}}

(or worse). A ( t 1 ) and h ′ ( t 1 ) are then chosen to remove this nonconforming rate, and in this way, we would obtain expressions for these functions and preserve the convergence rate.…”

Section: Perturbation Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multiscaling analysis of a slowly varying single species population model displaying an Allee effect

Idlango

Shepherd

Gear

2013

Math Methods in App Sciences

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Transitions in a logistic population model incorporating an Allee effect

Idlango

Gear

Shepherd

2015

Math Methods in App Sciences

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In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations.

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