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     * [10]Abstract
     * [11]Introduction
     * [12]Section snippets
     * [13]References (18)
     * [14]Cited by (9)

   [15]Elsevier

[16]Journal of Theoretical Biology

   [17]Volume 370, 7 April 2015, Pages 21-26
   [18]Journal of Theoretical Biology

L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge

   Author links open overlay panel (BUTTON) Vlastimil Krivan ^a ^b, (BUTTON) Anupam
   Priyadarshi ^a
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   [19]https://doi.org/10.1016/j.jtbi.2015.01.021[20]Get rights and content

Highlights

     * o Predator and prey isoclines are estimated from classic Gause³s experiments
       with protists feeding on yeast.
     * o It is shown that an L-shaped function fits prey isocline well.
     * o Such a shape of prey isocline is in agreement with predator-prey population
       models with a prey refuge.
     * o Lotka-Volterra and Rosenzweig-MacArthur models either with or without a
       prey refuge are parameterized by experimental data.
     * o Among them the one which fits data best is the Rosenzweig-MacArthur
       predator-prey model with a prey refuge.

Abstract

   Predator and prey isoclines are estimated from data on yeast-protist population
   dynamics ([21]Gause et al., 1936). Regression analysis shows that the prey
   isocline is best fitted by an L-shaped function that has a vertical and a
   horizontal part. The predator isocline is vertical. This shape of isoclines
   corresponds with the Lotka-Volterra and the Rosenzweig-MacArthur predator-prey
   models that assume a prey refuge. These results further support the idea that a
   prey refuge changes the prey isocline of predator-prey models from a horizontal
   to an L-shaped curve. Such a shape of the prey isocline effectively bounds
   amplitude of predator-prey oscillations, thus promotes species coexistence.

Introduction

   Isoclines have played an important role to identify mechanisms that regulate
   predator-prey coexistence (Rosenzweig and MacArthur, 1963). For the
   Lotka-Volterra predator-prey model, the predator isocline is vertical and the
   prey isocline is horizontal which leads to neutral oscillations in prey and
   predator population abundance. More realistic models with prey negative density
   dependence, predator density dependence, or non-linear functional responses lead
   to non-linear or sloped isoclines that can either stabilize or destabilize
   predator-prey population dynamics. In their seminal work, Rosenzweig and
   MacArthur (1963) analyzed effects of isoclines on predator and prey coexistence.
   Using graphical analysis they showed that a prey isocline with a vertical segment
   effectively bounds maximal oscillations in prey population numbers. Similarly, a
   predator isocline with a horizontal segment bounds maximal oscillations in
   predator numbers. Theoretical models that predict such isoclines can arise due to
   optimal prey selection by predators, or prey use of a refuge (e.g., Rosenzweig
   and MacArthur, 1963, Rosenzweig, 1977, Fryxell and Lundberg, 1994, Krivan, 1998,
   van Baalen et al., 2001, Brown and Kotler, 2004, Krivan, 2007, Krivan, 2013).

   In an attempt to falsify the Lotka-Volterra predator-prey model Gause
   experimented with various predator-prey systems (Gause, 1934, Gause, 1935a, Gause
   et al., 1936). Fig. 39 in Gause (1934) (see also Gause, 1935a) shows population
   dynamics of protist Paramecium bursaria feeding on yeast Schizasaccharomyces
   pombe, and protist Paramecium aurelia feeding on yeast Saccharomyces exiguus.
   Based on these experiments Gause (1935a, p. 45) concluded that "Quite clearly
   periodic fluctuations of the Lotka-Volterra type occurred". However, using the
   same data, Rosenzweig (1977) estimated a hump shaped prey isocline and a vertical
   predator isocline. Such isoclines are predicted by the Rosenzweig-MacArthur
   predator-prey model (Rosenzweig and MacArthur, 1963) that assumes a negative
   density dependent prey population growth and the Holling type II functional
   response. In this model predator-prey population dynamics can be destabilized by
   the "Paradox of Enrichment" when the environmental carrying capacity increases
   above a critical threshold and predator-prey population densities converge to a
   limit cycle (Rosenzweig, 1971).

   To better understand whether predator-prey oscillations were of the neutral
   Lotka-Volterra type, Gause with his collaborators (Gause, 1935b, Gause et al.,
   1936) continued experiments with P. bursaria feeding on yeast S. exiguus. In
   contrast to previous experiments, there was no aeration that would prevent yeast
   to sediment at the bottom of the beaker. The authors observed that below a
   critical prey threshold density all prey (yeast) sedimented and were unavailable
   to predators (protists) that lived in the water column. Consequently they
   developed a predator-prey model with a prey refuge. Graphical analysis of this
   model predicted, in accordance with their experimental observations, coexistence
   of prey and predators along a limit cycle. This is perhaps the first example of a
   limit cycle in the literature on predator-prey population dynamics. Their model
   was analyzed in detail by Krivan (2011) who showed that large refuges stabilize
   population dynamics at an equilibrium while smaller refuges lead to periodic
   oscillation in predator and prey numbers along a limit cycle. One distinguished
   feature of this model is that the prey isocline is L-shaped.

   In this paper we are interested in verifying whether the data on protists feeding
   on yeast from Gause et al. (1936) predict an L-shaped prey isocline. First, we
   estimate the shape of isoclines directly from the experimental data. Second, we
   fit several predator-prey models to the experimental data and compare the
   predicted and estimated predator and prey isoclines.

Section snippets

Direct isocline estimation from data

   In this paper we use data on population dynamics of protists (P. bursaria,
   squares in Fig. 1) feeding on yeast (S. exiguus, dots in Fig. 1) from Table 3 in
   Gause et al. (1936). The data represent 19 population experiments that differ in
   the initial predator [number of individuals/0.5 cm^3] and prey [number of
   individuals/0.1 mm^3] densities. Each time series consists of triples
   [MATH: (ti,xi,yi), :MATH]
   [MATH: i=1,...,N
   :MATH]
   where t[i] denotes day, and x[i] (y[i]) is the prey (predator) density at time
   t[i]. The longest time series

Models

   Gause et al. (1936) generalized the Lotka-Volterra predator-prey model by
   replacing the linear functional response by a non-linear functional response,
   i.e.,
   [MATH: dxdt=rx-yf(x)dydt=(ef(x)-m)y. :MATH]
   Here x (y) denotes the prey (predator) density, r is the per capita intrinsic
   prey population growth rate, e is the efficiency rate with which the captured
   prey are converted to new predators, and m is the predator mortality rate. In
   particular, Gause et al. (1936) assumed that (i) f is a saturating function,

Results

   Parameters for models were estimated using function NonlinearModelFit of
   Mathematica 10. The results are summarized in Table 1.

   We consider five models. Models 1 and 2 are the classical Lotka-Volterra
   predator-prey models (1) without any refuge and zero handling times (i.e., we set
   [MATH: xc=h=0 :MATH]
   in functional response (2)). In Model 1 we set r=0.5 and m=0.4 that are the
   estimates for the per capita intrinsic prey population growth rate and the
   predator mortality rate taken from Gause et al. (1936, p.

Discussion

   In this paper we estimate predator and prey isoclines using classic data on
   protists feeding on yeast (Gause et al., 1936). First, we estimate isoclines
   directly from data. These estimates suggest that the prey isocline is an L-shaped
   piece-wise linear line that has a horizontal and a vertical part (Fig. 2A, solid
   line) while the predator isocline is a vertical line in the prey-predator phase
   space (Fig. 2A, dashed line). Second, we parametrize the Lotka-Volterra and the
   Rosenzweig-MacArthur

Acknowledgments

   The suggestions from two anonymous reviewers and Burt Kotler are appreciated for
   improvements in the original version of the paper. Support provided by the
   Institute of Entomology (RVO:60077344) and Postdok-BIOGLOBE
   (CZ.1.07/2.3.00/30.0032) co-financed by the European Social Fund and the state
   budget of the Czech Republic is acknowledged.
   (BUTTON) Recommended articles

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   There are more references available in the full text version of this article.

Cited by (9)

     *

[26]Bird feeding and biodiversity: The decline of the Willow Tit
       2023, Ecological Economics
       (BUTTON) Show abstract
       Monitoring data from the British Trust for Ornithology document a decline in
       United Kingdom (UK) Willow Tit (Poecile montanus kleinschmidti) abundance
       that has been abrupt enough to be characterized by a catastrophe theoretic
       model. The reasons for the decline are thought to be some combination of
       increased nest competition and predation by the beneficiaries of a
       bird-feeder food subsidy, and habitat loss. A statistically estimated
       predator-prey catastrophe model is useful in illustrating how habitat loss
       and species interactions could combine to cause a precipitous population
       decline, and showing that, if the UK Willow Tit population is to recover, it
       will likely require both reversals of habitat loss and reductions in food
       subsidies to its competitors and predators. To achieve the latter, humans may
       have to collectively introduce rules that reduce the private enjoyment of
       backyard bird feeding to preserve broader ecological integrity.
     *

[27]Trend prey predator model - Analysis of gause model
       2019, Global Ecology and Conservation
       Citation Excerpt :
       Unfortunately, if just one component is ignored then the results can be
       totally misleading. This is the key reason why PP models have received
       considerable attention in scientific literature, see e. g. (Krivan and
       Priyadarshi, 2015; Zhang and Shen, 2015). A set of ordinary and/or partial
       nonlinear differential equations are frequently used descriptions of unsteady
       state behaviours of many different PP systems e. g. (Martín-Fernández et al.,
       2014; Vanegas-Acosta and Garzon-Alvarado, 2014).
       (BUTTON) Show abstract
       Any application of PP (Prey - Predator) models based on nonlinear
       differential equations requires identification of numerical values of all
       constants. This is often a problem because of severe information shortages.
       Many PP models are numerically sensitive and/or chaotic. Moreover, complex PP
       tasks are based on integration of differential equations with (partially)
       unknown numerical values of relevant constants and vague heuristics, e.g.
       vaguely described capture rate. These are the main reasons why PP numerical
       simulations cannot identify all important/relevant features, e.g. attractors.
       Trend models use just three values namely positive (increasing), zero
       (constant), negative (decreasing). A multiplication of a trend variable X by
       a positive constant a is irrelevant, it means that aX=( + )X=X. This obvious
       equation is used to eliminate all positive multiplicative constants a from PP
       mathematical models. A solution of a trend model is represented by a set of
       scenarios and a set of time transitions among these scenarios. A trend
       analogy of a quantitative phase portrait is represented by a discrete and
       finite set of scenarios and transitions. A trend version of the well-known
       Gause PP model is studied in details. The provably complete set of 41
       scenarios and 168 time transitions among them are given.
     *

[28]Do prey handling predators really matter: Subtle effects of a Crowley-Martin
functional response
       2017, Chaos, Solitons and Fractals
       Citation Excerpt :
       Thus it would be very interesting to look at spatially dependent interference
       coefficients (both for handling and searching predators) in patchy
       landscapes, or even at ODE models assuming patchy structure. To this end one
       might adopt the technology put forth in [15,20,32,36], as concerns patch
       dynamics, and prey refuges. Such endeavors would also be useful to better
       understand invasive species control, in patchy environments [13,20].
       (BUTTON) Show abstract
       Predator interference, or a decline in the per predator consumption rate as
       predator density increases, is generally considered a stabilizing mechanism
       in two-species predator-prey models. There is significant debate, as to
       whether prey handling predators, might interfere in the hunting process of
       prey searching predators, or whether these are mutually exclusive events. In
       the current manuscript, a three species food chain model, with strong top
       predator interference is considered. We prove that in terms of explosive
       instability/finite time blow up, sufficient interference by prey handling
       predators always tends to destabilize the system. The dynamics of a time
       delayed version, as well as the spatially explicit model are also explored.
       We use our results to comment on a certain paradox in ecological theory, as
       well as provide further insight into the nature of predator interference, and
       exploding populations of invasive species.
     *

[29]Modelling the interaction between a pest (Aculops lycopersici), two predators
(Pronematus ubiquitus and Macrolophus pygmaeus) and climate variables: a 3-year
greenhouse study in a tomato crop
       2023, Pest Management Science
     *

[30]Simplified modelling enhances biocontrol decision making in tomato greenhouses for
three important pest species
       2021, Journal of Pest Science
     *

[31]THE EFFECTS of INVASIVE EPIBIONTS on CRAB-MUSSEL COMMUNITIES: A THEORETICAL
APPROACH to UNDERSTAND MUSSEL POPULATION DECLINE
       2020, Journal of Biological Systems

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