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Cushing JM. A bifurcation theorem for Darwinian matrix models and an application to the evolution of reproductive life-history strategies. JOURNAL OF BIOLOGICAL DYNAMICS 2021; 15:S190-S213. [PMID: 33295254 DOI: 10.1080/17513758.2020.1858196] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2020] [Accepted: 11/20/2020] [Indexed: 06/12/2023]
Abstract
We prove bifurcation theorems for evolutionary game theoretic (Darwinian dynamic) versions of nonlinear matrix equations for structured population dynamics. These theorems generalize existing theorems concerning the bifurcation and stability of equilibrium solutions when an extinction equilibrium destabilizes by allowing for the general appearance of the bifurcation parameter. We apply the theorems to a Darwinian model designed to investigate the evolutionary selection of reproductive strategies that involve either low or high post-reproductive survival (semelparity or iteroparity). The model incorporates the phenotypic trait dependence of two features: population density effects on fertility and a trade-off between inherent fertility and post-reproductive survival. Our analysis of the model determines conditions under which evolution selects for low or for high reproductive survival. In some cases (notably an Allee component effect on newborn survival), the model predicts multiple attractor scenarios in which low or high reproductive survival is initial condition dependent.
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Affiliation(s)
- J M Cushing
- Department of Mathematics, Interdisciplinary Program in Applied Mathematics, University of Arizona, Tucson, AZ, USA
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CUSHING JM, STEFANKO KATHRYN. A DARWINIAN DYNAMIC MODEL FOR THE EVOLUTION OF POST-REPRODUCTION SURVIVAL. J BIOL SYST 2021. [DOI: 10.1142/s0218339021400088] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
We derive and study a Darwinian dynamic model based on a low-dimensional discrete- time population model focused on two features: density-dependent fertility and a trade-off between inherent (density free) fertility and post-reproduction survival. Both features are assumed to be dependent on a phenotypic trait subject to natural selection. The model tracks the dynamics of the population coupled with that of the population mean trait. We study the stability properties of equilibria by means of bifurcation theory. Whether post-reproduction survival at equilibrium is low or high is shown, in this model, to depend significantly on the nature of the trait dependence of the density effects. An Allee effect can also play a significant role.
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Affiliation(s)
- J. M. CUSHING
- Department of Mathematics and Interdisciplinary, Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721, USA
| | - KATHRYN STEFANKO
- Department of Mathematics and Interdisciplinary, Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721, USA
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Cushing JM. Difference equations as models of evolutionary population dynamics. JOURNAL OF BIOLOGICAL DYNAMICS 2019; 13:103-127. [PMID: 30714512 DOI: 10.1080/17513758.2019.1574034] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2018] [Accepted: 01/21/2019] [Indexed: 06/09/2023]
Abstract
We describe the evolutionary game theoretic methodology for extending a difference equation population dynamic model in a way so as to account for the Darwinian evolution of model coefficients. We give a general theorem that describes the familiar transcritical bifurcation that occurs in non-evolutionary models when theextinction equilibrium destabilizes. This bifurcation results in survival (positive) equilibria whose stability depends on the direction of bifurcation. We give several applications based on evolutionary versions of some classic equations, such as the discrete logistic (Beverton-Holt) and Ricker equations. In addition to illustrating our theorems, these examples also illustrate other biological phenomena, such as strong Allee effects, time-dependent adaptive landscapes, and evolutionary stable strategies.
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Affiliation(s)
- J M Cushing
- a Department of Mathematics, Interdisciplinary Program in Applied Mathematics , University of Arizona , Tucson, AZ , USA
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Elaydi S, Kwessi E, Livadiotis G. Hierarchical competition models with the Allee effect III: multispecies. JOURNAL OF BIOLOGICAL DYNAMICS 2018; 12:271-287. [PMID: 29508637 DOI: 10.1080/17513758.2018.1439537] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/17/2017] [Accepted: 02/05/2018] [Indexed: 06/08/2023]
Abstract
A general notion of the Allee effect for higher dimensional triangular maps is proposed. A global dynamics theory is established. The theory is applied to multi-species hierarchical models. Then we provide a detailed study of the global dynamics of three-species Ricker competition models with the Allee effect. Regions of extinction, exclusion and coexistence are identified.
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Affiliation(s)
- S Elaydi
- a Trinity University , San Antonio , TX , USA
| | - E Kwessi
- a Trinity University , San Antonio , TX , USA
| | - G Livadiotis
- b Southwest Research Institute , San Antonio , TX , USA
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Marvá M, San Segundo F. Age-structure density-dependent fertility and individuals dispersal in a population model. Math Biosci 2018; 300:157-167. [PMID: 29608888 DOI: 10.1016/j.mbs.2018.03.029] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2017] [Revised: 02/20/2018] [Accepted: 03/29/2018] [Indexed: 10/17/2022]
Abstract
In this work, we analyze the interplay between general age structured density-dependent fertility functions and age classes dispersal in a patchy environment. As novelties, (i) the fertility function depends on age classes (instead of on the total population size) and (ii) dispersal patterns are also allowed to be different for individuals belonging to different age classes. Our results highlight the interplay between the shape of the age structured density-dependent fertility function and the age classes dispersal patterns. We analyze this interaction from an environmental management point of view by exploring the consequences of connecting patches that can sustain a population (source patch) or cannot (sink patch), as well as its relation to component Allee effects and strong Allee effects. In particular, we have found scenarios such that the metapopulation goes extinct when two isolated source patches are connect due to heterogeneous age classes distribution. On the contrary, there are settings such that heterogeneous age classes distribution enables two isolated sink patches to be sustainable when connected. Besides, we discuss what kind of local interventions are helpful to manage component Allee effect and its impact at the metopopulation level. The source code used to simulations is fully available. The code is presented as a knitr reproducible document in the open source R computing system. Thus, free access and usability of the code are granted.
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Affiliation(s)
- M Marvá
- U. D. Matemáticas, Universidad de Alcalá, Alcalá de Henares 28871, Spain.
| | - F San Segundo
- U. D. Matemáticas, Universidad de Alcalá, Alcalá de Henares 28871, Spain.
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Cushing JM, Henson SM. Periodic matrix models for seasonal dynamics of structured populations with application to a seabird population. J Math Biol 2018; 77:1689-1720. [PMID: 29397422 DOI: 10.1007/s00285-018-1211-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2017] [Revised: 01/06/2018] [Indexed: 11/29/2022]
Abstract
For structured populations with an annual breeding season, life-stage interactions and behavioral tactics may occur on a faster time scale than that of population dynamics. Motivated by recent field studies of the effect of rising sea surface temperature (SST) on within-breeding-season behaviors in colonial seabirds, we formulate and analyze a general class of discrete-time matrix models designed to account for changes in behavioral tactics within the breeding season and their dynamic consequences at the population level across breeding seasons. As a specific example, we focus on egg cannibalism and the daily reproductive synchrony observed in seabirds. Using the model, we investigate circumstances under which these life history tactics can be beneficial or non-beneficial at the population level in light of the expected continued rise in SST. Using bifurcation theoretic techniques, we study the nature of non-extinction, seasonal cycles as a function of environmental resource availability as they are created upon destabilization of the extinction state. Of particular interest are backward bifurcations in that they typically create strong Allee effects in population models which, in turn, lead to the benefit of possible (initial condition dependent) survival in adverse environments. We find that positive density effects (component Allee effects) due to increased adult survival from cannibalism and the propensity of females to synchronize daily egg laying can produce a strong Allee effect due to a backward bifurcation.
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Affiliation(s)
- J M Cushing
- Department of Mathematics and Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 84721, USA.
| | - Shandelle M Henson
- Department of Mathematics, Andrews University, 4260 Administration Drive, Berrien Springs, MI, 49104, USA.,Department of Biology, Andrews University, 4280 Administration Drive, Berrien Springs, MI, 49104, USA
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On the intrinsic dynamics of bacteria in waterborne infections. Math Biosci 2017; 296:71-81. [PMID: 29291431 DOI: 10.1016/j.mbs.2017.12.005] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2017] [Revised: 12/02/2017] [Accepted: 12/26/2017] [Indexed: 11/23/2022]
Abstract
The intrinsic dynamics of bacteria often play an important role in the transmission and spread of waterborne infectious diseases. In this paper, we construct mathematical models for waterborne infections and analyze two types of nontrivial bacterial dynamics: logistic growth, and growth with Allee effects. For the model with logistic growth, we find that regular threshold dynamics take place, and the basic reproduction number can be used to characterize disease extinction and persistence. In contrast, the model with Allee effects exhibits much more complex dynamics, including the existence of multiple endemic equilibria and the presence of backward bifurcation and forward hysteresis.
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Veprauskas A, Cushing JM. A juvenile-adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects. JOURNAL OF BIOLOGICAL DYNAMICS 2017; 11:1-24. [PMID: 26840043 DOI: 10.1080/17513758.2015.1131853] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We study a discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull (Larus glaucescens). The model focuses on mechanisms hypothesized to play key roles in a population's response to degraded environment resources, namely, increased cannibalism and adjustments in reproductive timing. We explore the dynamic consequences of these mechanics using a juvenile-adult structure model. Mathematically, the model is unusual in that it involves a high co-dimension bifurcation at [Formula: see text] which, in turn, leads to a dynamic dichotomy between equilibrium states and synchronized oscillatory states. We give diagnostic criteria that determine which dynamic is stable. We also explore strong Allee effects caused by positive feedback mechanisms in the model and the possible consequence that a cannibalistic population can survive when a non-cannibalistic population cannot.
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Affiliation(s)
- Amy Veprauskas
- a Interdisciplinary Program in Applied Mathematics , University of Arizona , Tucson , AZ , USA
| | - J M Cushing
- a Interdisciplinary Program in Applied Mathematics , University of Arizona , Tucson , AZ , USA
- b Department of Mathematics , University of Arizona , Tucson , AZ , USA
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Cushing JM, Martins F, Pinto AA, Veprauskas A. A bifurcation theorem for evolutionary matrix models with multiple traits. J Math Biol 2017; 75:491-520. [DOI: 10.1007/s00285-016-1091-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2016] [Revised: 10/03/2016] [Indexed: 11/25/2022]
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Sewalt L, Harley K, van Heijster P, Balasuriya S. Influences of Allee effects in the spreading of malignant tumours. J Theor Biol 2016; 394:77-92. [DOI: 10.1016/j.jtbi.2015.12.024] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2015] [Revised: 11/10/2015] [Accepted: 12/30/2015] [Indexed: 12/31/2022]
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Cushing JM. The evolutionary dynamics of a population model with a strong Allee effect. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2015; 12:643-660. [PMID: 25974340 DOI: 10.3934/mbe.2015.12.643] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).
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Affiliation(s)
- Jim M Cushing
- Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, Tucson, Arizona, 85721, United States.
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Assas L, Dennis B, Elaydi S, Kwessi E, Livadiotis G. Hierarchical competition models with the Allee effect II: the case of immigration. JOURNAL OF BIOLOGICAL DYNAMICS 2015; 9:288-316. [PMID: 26394840 DOI: 10.1080/17513758.2015.1077999] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
This is part II of an earlier paper that dealt with hierarchical models with the Allee effect but with no immigration. In this paper, we greatly simplify the proofs in part I and provide a proof of the global dynamics of the non-hyperbolic cases that were previously conjectured. Then, we show how immigration to one of the species or to both would, drastically, change the dynamics of the system. It is shown that if the level of immigration to one or to both species is above a specified level, then there will be no extinction region where both species go to extinction.
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Affiliation(s)
- Laila Assas
- a Department of Mathematics , King Abdul Aziz University , Saudi Arabia
| | - Brian Dennis
- b Departement of Mathematics , University of Idaho , 875 Perimeter Drive MS 1136 Moscow, ID 83844 , USA
| | - Saber Elaydi
- c Department of Fish and Wildlife Science , Trinity University, One Trinity Place , San Antonio , TX 78212 , USA
| | - Eddy Kwessi
- d Department of Mathematics , Trinity University , One Trinity Place, San Antonio , TX 78212 , USA
| | - George Livadiotis
- e Department of Space Science , Southwest Research Institute , San Antonio , TX 78238 , USA
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Assas L, Elaydi S, Kwessi E, Livadiotis G, Ribble D. Hierarchical competition models with Allee effects. JOURNAL OF BIOLOGICAL DYNAMICS 2014; 9 Suppl 1:32-44. [PMID: 24916355 DOI: 10.1080/17513758.2014.923118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
We consider a two-species hierarchical competition model with a strong Allee effect. The Allee effect is assumed to be caused by predator saturation. Moreover, we assume that there is a 'silverback' species x that gets first choice of the resources and where growth is limited by its own intraspecific competition, while the second 'inferior' species y gets whatever is left. Both species x and y are assumed to have the property of strong Allee effect. In this paper we determine the impact of the presence of the Allee effect on the global dynamics of both species.
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Affiliation(s)
- Laila Assas
- a Department of Mathematics , Umm Al-Qura University , B.O.X 10265, Makkah , Saudi Arabia
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