1
|
Nie H, Qin Q, Zhang L. Dynamics analysis of a reaction-diffusion-advection benthic-drift model with logistic growth. J Math Biol 2025; 90:23. [PMID: 39899044 DOI: 10.1007/s00285-025-02183-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2024] [Revised: 12/12/2024] [Accepted: 01/09/2025] [Indexed: 02/04/2025]
Abstract
This paper aims to investigate the benthic-drift population model in both open and closed advective environments, focusing on the logistic growth of benthic populations. We obtain the threshold dynamics using the monotone iteration method, and show that the zero solution is globally attractive straightforward when linearly stable. When unstable, limits from monotonic iteration of upper and lower solutions are upper and lower semi-continuous, respectively. By employing a part metric, we prove these limits are equal and continuous, leading to a positive steady state. In the critical case, we establish that the limit function from the upper solution iteration must be the zero solution by analyzing an algebraic equation. Furthermore, we conduct a quantitative analysis of the principal eigenvalue for a non-self-adjoint eigenvalue problem to examine how the diffusion rate, advection rate, and population release rates influence the dynamics. The results suggest that the diffusion rate and advection rate have distinct effects on population dynamics in open and closed advective environments, depending on the population release rates.
Collapse
Affiliation(s)
- Hua Nie
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, 710119, China
| | - Qian Qin
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, 710119, China
| | - Lei Zhang
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi, 710119, China.
| |
Collapse
|
2
|
Zhang J, Han X, Wang H. Phytoplankton-chytrid-zooplankton dynamics via a reaction-diffusion-advection mycoloop model. J Math Biol 2024; 89:15. [PMID: 38884837 DOI: 10.1007/s00285-024-02113-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2023] [Revised: 03/17/2024] [Accepted: 05/26/2024] [Indexed: 06/18/2024]
Abstract
Mycoloop is an important aquatic food web composed of phytoplankton, chytrids (one dominant group of parasites in aquatic ecosystems), and zooplankton. Chytrids infect phytoplankton and fragment them for easy consumption by zooplankton. The free-living chytrid zoospores are also a food resource for zooplankton. A dynamic reaction-diffusion-advection mycoloop model is proposed to describe the Phytoplankton-chytrid-zooplankton interactions in a poorly mixed aquatic environment. We analyze the dynamics of the mycoloop model to obtain dissipativity, steady state solutions, and persistence. We rigorously derive several critical thresholds for phytoplankton or zooplankton invasion and chytrid transmission among phytoplankton. Numerical diagrams show that varying ecological factors affect the formation and breakup of the mycoloop, and zooplankton can inhibit chytrid transmission among phytoplankton. Furthermore, this study suggests that mycoloop may either control or cause phytoplankton blooms.
Collapse
Affiliation(s)
- Jimin Zhang
- School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People's Republic of China
| | - Xu Han
- School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People's Republic of China
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada.
| |
Collapse
|
3
|
Zhang J, Yan Y, Shi J. Modelling phytoplankton-virus interactions: phytoplankton blooms and lytic virus transmission. J Math Biol 2024; 88:77. [PMID: 38695878 PMCID: PMC11065791 DOI: 10.1007/s00285-024-02093-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2023] [Revised: 01/17/2024] [Accepted: 04/08/2024] [Indexed: 05/05/2024]
Abstract
A dynamic reaction-diffusion model of four variables is proposed to describe the spread of lytic viruses among phytoplankton in a poorly mixed aquatic environment. The basic ecological reproductive index for phytoplankton invasion and the basic reproduction number for virus transmission are derived to characterize the phytoplankton growth and virus transmission dynamics. The theoretical and numerical results from the model show that the spread of lytic viruses effectively controls phytoplankton blooms. This validates the observations and experimental results of Emiliana huxleyi-lytic virus interactions. The studies also indicate that the lytic virus transmission cannot occur in a low-light or oligotrophic aquatic environment.
Collapse
Affiliation(s)
- Jimin Zhang
- School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People's Republic of China
| | - Yawen Yan
- School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, Heilongjiang, People's Republic of China
| | - Junping Shi
- Department of Mathematics, William & Mary, Williamsburg, VA, 23187-8795, USA.
| |
Collapse
|
4
|
Nguyen TD, Wu Y, Tang T, Veprauskas A, Zhou Y, Rouhani BD, Shuai Z. Impact of resource distributions on the competition of species in stream environment. J Math Biol 2023; 87:62. [PMID: 37736867 DOI: 10.1007/s00285-023-01978-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2023] [Revised: 07/27/2023] [Accepted: 07/31/2023] [Indexed: 09/23/2023]
Abstract
Our earlier work in Nguyen et al. (Maximizing metapopulation growth rate and biomass in stream networks. arXiv preprint arXiv:2306.05555 , 2023) shows that concentrating resources on the upstream end tends to maximize the total biomass in a metapopulation model for a stream species. In this paper, we continue our research direction by further considering a Lotka-Volterra competition patch model for two stream species. We show that the species whose resource allocations maximize the total biomass has the competitive advantage.
Collapse
Affiliation(s)
- Tung D Nguyen
- Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA
| | - Yixiang Wu
- Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, 37132, USA.
| | - Tingting Tang
- Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182, USA
| | - Amy Veprauskas
- Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70501, USA
| | - Ying Zhou
- Department of Mathematics, Lafayette College, Easton, PA, 18042, USA
| | - Behzad Djafari Rouhani
- Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, 79968, USA
| | - Zhisheng Shuai
- Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA
| |
Collapse
|
5
|
Cosner C, Rodríguez N. On the Allee effect and directed movement on the whole space. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:8010-8030. [PMID: 37161183 DOI: 10.3934/mbe.2023347] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/11/2023]
Abstract
It is well known that relocation strategies in ecology can make the difference between extinction and persistence. We consider a reaction-advection-diffusion framework to analyze movement strategies in the context of species which are subject to a strong Allee effect. The movement strategies we consider are a combination of random Brownian motion and directed movement through the use of an environmental signal. We prove that a population can overcome the strong Allee effect when the signals are super-harmonic. In other words, an initially small population can survive in the long term if they aggregate sufficiently fast. A sharp result is provided for a specific signal that can be related to the Fokker-Planck equation for the Orstein-Uhlenbeck process. We also explore the case of pure diffusion and pure aggregation and discuss their benefits and drawbacks, making the case for a suitable combination of the two as a better strategy.
Collapse
Affiliation(s)
- Chris Cosner
- University of Miami, Department of Mathematics, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, USA
| | - Nancy Rodríguez
- CU Boulder, Department of Applied Mathematics, 11 Engineering Dr, Boulder, CO 80309, USA
| |
Collapse
|
6
|
Lou Y, Nie H. Global dynamics of a generalist predator-prey model in open advective environments. J Math Biol 2022; 84:46. [PMID: 35499718 DOI: 10.1007/s00285-022-01756-w] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2021] [Revised: 04/06/2022] [Accepted: 04/06/2022] [Indexed: 10/18/2022]
Abstract
This paper deals with a system of reaction-diffusion-advection equations for a generalist predator-prey model in open advective environments, subject to an unidirectional flow. In contrast to the specialist predator-prey model, the dynamics of this system is more complex. It turns out that there exist some critical advection rates and predation rates, which classify the global dynamics of the generalist predator-prey system into three or four scenarios: (1) coexistence; (2) persistence of prey only; (3) persistence of predators only; and (4) extinction of both species. Moreover, the results reveal significant differences between the specialist predator-prey system and the generalist predator-prey system, including the evolution of the critical predation rates with respect to the ratio of the flow speeds; the take-over of the generalist predator; and the reduction in parameter range for the persistence of prey species alone. These findings may have important biological implications on the invasion of generalist predators in open advective environments.
Collapse
Affiliation(s)
- Yuan Lou
- School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, China
| | - Hua Nie
- School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, 710119, Shaanxi, China.
| |
Collapse
|
7
|
Warne DJ, Baker RE, Simpson MJ. Rapid Bayesian Inference for Expensive Stochastic Models. J Comput Graph Stat 2021. [DOI: 10.1080/10618600.2021.2000419] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- David J. Warne
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| | - Ruth E. Baker
- Mathematical Institute, University of Oxford, Oxford, UK
| | - Matthew J. Simpson
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
| |
Collapse
|
8
|
Jiang H, Lam KY, Lou Y. Three-patch Models for the Evolution of Dispersal in Advective Environments: Varying Drift and Network Topology. Bull Math Biol 2021; 83:109. [PMID: 34524555 DOI: 10.1007/s11538-021-00939-8] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Accepted: 08/18/2021] [Indexed: 10/20/2022]
Abstract
We study the evolution of dispersal in advective three-patch models with distinct network topologies. Organisms can move between connected patches freely and they are also subject to the passive, directed drift. The carrying capacity is assumed to be the same in all patches, while the drift rates could vary. We first show that if all drift rates are the same, the faster dispersal rate is selected for all three models. For general drift rates, we show that the infinite diffusion rate is a local Convergence Stable Strategy (CvSS) for all three models. However, there are notable differences for three models: For Model I, the faster dispersal is always favored, irrespective of the drift rates, and thus the infinity dispersal rate is a global CvSS. In contrast, for Models II and III a singular strategy will exist for some ranges of drift rates and bi-stability phenomenon happens, i.e., both infinity and zero diffusion rates are local CvSSs. Furthermore, for both Models II and III, it is possible for two competing populations to coexist by varying drift and diffusion rates. Some predictions on the dynamics of n-patch models in advective environments are given along with some numerical evidence.
Collapse
Affiliation(s)
- Hongyan Jiang
- Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China
| | - King-Yeung Lam
- Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA.
| | - Yuan Lou
- School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China.,Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA
| |
Collapse
|
9
|
Rana S, Bhowmick AR, Sardar T. Invasive dynamics for a predator-prey system with Allee effect in both populations and a special emphasis on predator mortality. CHAOS (WOODBURY, N.Y.) 2021; 31:033150. [PMID: 33810739 DOI: 10.1063/5.0035566] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2020] [Accepted: 03/09/2021] [Indexed: 06/12/2023]
Abstract
We considered a non-linear predator-prey model with an Allee effect on both populations on a two spatial dimension reaction-diffusion setup. Special importance to predator mortality was given as it may be often controlled through human-made harvesting processes. The local dynamics of the model was studied through boundedness, equilibrium, and stability analysis. An extensive numerical stability analysis was performed and found that bi-stability is not possible for the non-spatial model. By analyzing the spatial model, we found the condition for successful invasion and the persistence region of the species based on the predator Allee effect and its mortality parameter. Four different dynamics in this region of the parameter space are mainly explored. First, the Allee effect on both populations leads to various new types of species spread. Second, for a high value of per-capita growth rate, two completely new spreads (e.g., sun surface, colonial) have been found depending on the Allee effect parameter. Third, the Allee coefficient on the predator population leads to spatiotemporal chaos via a patchy spread for both linear and quadratic mortality rates. Finally, a more rigorous analysis is performed to study the chaotic nature of the system within the whole persistence domain. We have studied the possibility of chaos through temporal variation in different invasion regions. Furthermore, the chaotic fluctuation is studied through the sensitivity of initial conditions and by investigating the dominant Lyapunov exponent value.
Collapse
Affiliation(s)
- Sourav Rana
- Department of Statistics, Visva-Bharati University, Santiniketan 731235, India
| | | | - Tridip Sardar
- Department of Mathematics, Dinabandhu Andrews College, Kolkata 700084, India
| |
Collapse
|
10
|
Travelling wave solutions in a negative nonlinear diffusion-reaction model. J Math Biol 2020; 81:1495-1522. [PMID: 33216181 PMCID: PMC7717045 DOI: 10.1007/s00285-020-01547-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/21/2019] [Revised: 02/04/2020] [Accepted: 08/22/2020] [Indexed: 12/20/2022]
Abstract
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c^*$$\end{document}c∗, and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.
Collapse
|
11
|
Chang X, Shi J, Wang H. Spatial modeling and dynamics of organic matter biodegradation in the absence or presence of bacterivorous grazing. Math Biosci 2020; 331:108501. [PMID: 33166582 DOI: 10.1016/j.mbs.2020.108501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2020] [Revised: 09/29/2020] [Accepted: 10/26/2020] [Indexed: 10/23/2022]
Abstract
Biodegradation is a pivotal natural process for elemental recycling and preservation of an ecosystem. Mechanistic modeling of biodegradation has to keep track of chemical elements via stoichiometric theory, under which we propose and analyze a spatial movement model in the absence or presence of bacterivorous grazing. Sensitivity analysis shows that the organic matter degradation rate is most sensitive to the grazer's death rate when the grazer is present and most sensitive to the bacterial death rate when the grazer is absent. Therefore, these two death rates are chosen as the primary parameters in the conditions of most mathematical theorems. The existence, stability and persistence of solutions are proven by applying linear stability analysis, local and global bifurcation theory, and the abstract persistence theory. Through numerical simulations, we obtain the transient and asymptotic dynamics and explore the effects of all parameters on the organic matter decomposition. Grazers either facilitate biodegradation or has no impact on biodegradation, which resolves the "decomposition-facilitation paradox" in the spatial context.
Collapse
Affiliation(s)
- Xiaoyuan Chang
- Department of Mathematics, Harbin University of Science and Technology, Harbin, Heilongjiang, 150080, PR China
| | - Junping Shi
- Department of Mathematics, William & Mary, Williamsburg, VA, 23187-8795, USA.
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
| |
Collapse
|
12
|
Jiang H, Lam KY, Lou Y. Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules. Bull Math Biol 2020; 82:131. [PMID: 33025293 DOI: 10.1007/s11538-020-00803-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2020] [Accepted: 09/07/2020] [Indexed: 11/25/2022]
Abstract
We study the dynamics of two competing species in three-patch models and illustrate how the topology of directed river network modules may affect the evolution of dispersal. Each model assumes that patch 1 is at the upstream end, patch 3 is at the downstream end, but patch 2 could be upstream, or middle stream, or downstream, depending on the specific topology of the modules. We posit that individuals are subject to both unbiased dispersal between patches and passive drift from one patch to another, depending upon the connectivity of patches. When the drift rate is small, we show that for all models, the mutant species can invade when rare if and only if it is the slower disperser. However, when the drift rate is large, most models predict that the faster disperser wins, while some predict that there exists one evolutionarily singular strategy. The intermediate range of drift is much more complex: most models predict the existence of one singular strategy, but it may or may not be evolutionarily stable, again depending upon the topology of modules, while one model even predicts that for some intermediate drift rate, singular strategy does not exist and the faster disperser wins the competition.
Collapse
Affiliation(s)
- Hongyan Jiang
- Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China
| | - King-Yeung Lam
- Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA
| | - Yuan Lou
- Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA.
| |
Collapse
|