Secondary Bifurcations in a Lotka–Volterra Model for N Competitors with Applications to Action Selection and Compulsive Behaviors

2014 ◽  
Vol 24 (12) ◽  
pp. 1450156 ◽  
Author(s):  
T. D. Frank
Keyword(s):  
Fixed Points ◽  
Growth Rates ◽  
Exponential Growth ◽  
Action Selection ◽  
Volterra Model ◽  
Selection Principle ◽  
Compulsive Behaviors ◽  
Winner Takes All ◽  
Secondary Bifurcations ◽  
Cross Inhibition

A Lotka–Volterra model for an arbitrary number of competitors is studied for different ratios of self-inhibition versus cross-inhibition. It is shown that winner-takes-all fixed points (states of single surviving species) are the only stable fixed points of the model when cross-inhibition exceeds self-inhibition. Secondary bifurcations in terms of bifurcations between winner-takes-all fixed points induced by changes in the exponential growth rates of competitors are studied and the critical control parameters are identified. A selection principle is derived that states that evolution proceeds in such a way that exponential growth rates of surviving competitors are magnified in evolutionary bifurcation steps. The interacting competitor model is applied as an amplitude equation model for interacting patterns of self-organizing pattern formation systems with an eye on action selection and compulsive behaviors in humans. The possibility is discussed that human behavior is subjected to the selection principle of "faster growth rates".

A multitype decomposable age-dependent branching process and its applications

1995 ◽  
Vol 32 (3) ◽  
pp. 591-608 ◽  
Author(s):  
Chinsan Lee ◽  
Grace L. Yang
Keyword(s):  
Tumor Growth ◽  
Life Span ◽  
Growth Rates ◽  
Exponential Growth ◽  
Branching Process ◽  
Asymptotic Formulas ◽  
Stage Model ◽  
Two Stage ◽  
Markov Branching Process ◽  
Age Dependent

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.

scholarly journals On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

2020 ◽  
Vol 61 ◽  
pp. C1-C14
Author(s):  
Hidekazu Yoshioka ◽  
Yumi Yoshioka
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Growth Rates ◽  
Epidemic Model ◽  
Volterra Model ◽  
Sis Epidemic Model ◽  
Sis Epidemic

We propose and analyse a new stochastic competing two-species population dynamics model. Competing algae population dynamics in river environments, an important engineering problem, motivates this model. The algae dynamics are described by a system of stochastic differential equations with the characteristic that the two populations are competing with each other through the environmental capacities. Unique existence of the uniformly bounded strong solution is proven and an attractor is identified. The Kolmogorov backward equation associated with the population dynamics is formulated and its unique solvability in a Banach space with a weighted norm is discussed. Our mathematical analysis results can be effectively utilized for a foundation of modelling, analysis, and control of the competing algae population dynamics. References S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two correlated brownian motions. Nonlin. Dyn., 97(4):2175–2187, 2019. doi:10.1007/s11071-019-05114-2. S. Cai, Y. Cai, and X. Mao. A stochastic differential equation SIS epidemic model with two independent brownian motions. J. Math. Anal. App., 474(2):1536–1550, 2019. doi:10.1016/j.jmaa.2019.02.039. U. Callies, M. Scharfe, and M. Ratto. Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe river. Ecol. Mod., 213(2):229–244, 2008. doi:10.1016/j.ecolmodel.2007.12.015. M. G. Crandall, H. Ishii, and P. L. Lions. User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc., 27(1):229–244, 1992. doi:10.1090/S0273-0979-1992-00266-5. N. H. Du and V. H. Sam. Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J. Math. Anal. App., 324(1):82–97, 2006. doi:10.1016/j.jmaa.2005.11.064. P. Grandits, R. M. Kovacevic, and V. M. Veliov. Optimal control and the value of information for a stochastic epidemiological SIS model. J. Math. Anal. App., 476(2):665–695, 2019. doi:10.1016/j.jmaa.2019.04.005. B. Horvath and O. Reichmann. Dirichlet forms and finite element methods for the SABR model. SIAM J. Fin. Math., 9(2):716–754, 2018. doi:10.1137/16M1066117. J. Hozman and T. Tichy. DG framework for pricing european options under one-factor stochastic volatility models. J. Comput. Appl. Math., 344:585–600, 2018. doi:10.1016/j.cam.2018.05.064. G. Lan, Y. Huang, C. Wei, and S. Zhang. A stochastic SIS epidemic model with saturating contact rate. Physica A, 529(121504):1–14, 2019. doi:10.1016/j.physa.2019.121504. J. L. Lions and E. Magenes. Non-homogeneous Boundary Value Problems and Applications (Vol. 1). Springer Berlin Heidelberg, 1972. doi:10.1007/978-3-642-65161-8. J. Lv, X. Zou, and L. Tian. A geometric method for asymptotic properties of the stochastic Lotka–Volterra model. Commun. Nonlin. Sci. Numer. Sim., 67:449–459, 2019. doi:10.1016/j.cnsns.2018.06.031. S. Morin, M. Coste, and F. Delmas. A comparison of specific growth rates of periphytic diatoms of varying cell size under laboratory and field conditions. Hydrobiologia, 614(1):285–297, 2008. doi:10.1007/s10750-008-9513-y. B. \T1\O ksendal. Stochastic Differential Equations. Springer Berlin Heidelberg, 2003. doi:10.1007/978-3-642-14394-6. O. Oleinik and E. V. Radkevic. Second-order Equations with Nonnegative Characteristic Form. Springer Boston, 1973. doi:10.1007/978-1-4684-8965-1. S. Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty: with Robust CLT and G-Brownian Motion. Springer-Verlag Berlin Heidelberg, 2019. doi:10.1007/978-3-662-59903-7. T. S. Schmidt, C. P. Konrad, J. L. Miller, S. D. Whitlock, and C. A. Stricker. Benthic algal (periphyton) growth rates in response to nitrogen and phosphorus: parameter estimation for water quality models. J. Am. Water Res. Ass., 2019. doi:10.1111/1752-1688.12797. Y. Toda and T. Tsujimoto. Numerical modeling of interspecific competition between filamentous and nonfilamentous periphyton on a flat channel bed. Landscape Ecol. Eng., 6(1):81–88, 2010. doi:10.1007/s11355-009-0093-4. H. Yoshioka, Y. Yaegashi, Y. Yoshioka, and K. Tsugihashi. Optimal harvesting policy of an inland fishery resource under incomplete information. Appl. Stoch. Models Bus. Ind., 35(4):939–962, 2019. doi:10.1002/asmb.2428.

scholarly journals High temperature decreases the PIC / POC ratio and increases phosphorus requirements in <i>Coccolithus pelagicus</i> (Haptophyta)

2014 ◽  
Vol 11 (1) ◽  
pp. 1021-1051 ◽  
Author(s):  
A. C. Gerecht ◽  
L. Šupraha ◽  
B. Edvardsen ◽  
I. Probert ◽  
J. Henderiks
Keyword(s):  
Elevated Temperature ◽  
Growth Rates ◽  
Exponential Growth ◽  
Phosphorus Limitation ◽  
Photic Zone ◽  
P Limitation ◽  
P Content ◽  
Phytoplankton Communities ◽  
Global Biogeochemical Cycles ◽  
Coccolithus Pelagicus

Abstract. Rising ocean temperatures will likely increase stratification of the water column and reduce nutrient input into the photic zone. This will increase the likelihood of nutrient limitation in marine microalgae, leading to changes in the abundance and composition of phytoplankton communities, which in turn will affect global biogeochemical cycles. Calcifying algae, such as coccolithophores, influence the carbon cycle by fixing CO2 into particulate organic carbon (POC) through photosynthesis and into particulate inorganic carbon (PIC) through calcification. As calcification produces a net release of CO2, the ratio of PIC / POC determines whether coccolithophores act as a source (PIC / POC > 1) or a sink (PIC / POC < 1) of atmospheric CO2. We studied the effect of phosphorus (P-) limitation and temperature stress on the physiology and PIC / POC ratios of two subspecies of Coccolithus pelagicus. This large and heavily calcified species (PIC / POC generally > 1.5) is a major contributor to calcite export from the photic zone into deep-sea reservoirs. Phosphorus limitation did not influence exponential growth rates in either subspecies, but P-limited cells had significantly lower cellular P-content. A 5 °C temperature increase did not affect exponential growth rates either, but nearly doubled cellular P-content under both high and low phosphate availability. The PIC / POC ratios did not differ between P-limited and nutrient-replete cultures, but at elevated temperature (from 10 to 15 °C) PIC / POC ratios decreased by 40–60%. Our results suggest that elevated temperature may intensify P-limitation due to a higher P-requirement to maintain growth and POC production rates, possibly reducing abundances in a warmer ocean. Under such a scenario C. pelagicus may decrease its calcification rate relative to photosynthesis, resulting in PIC / POC ratios < 1 and favouring CO2-sequestration over release. Phosphorus limitation by itself is unlikely to cause changes in the PIC / POC ratio in this species.

scholarly journals High temperature decreases the PIC / POC ratio and increases phosphorus requirements in <i>Coccolithus pelagicus</i> (Haptophyta)

2014 ◽  
Vol 11 (13) ◽  
pp. 3531-3545 ◽  
Author(s):  
A. C. Gerecht ◽  
L. Šupraha ◽  
B. Edvardsen ◽  
I. Probert ◽  
J. Henderiks
Keyword(s):  
High Temperature ◽  
Growth Rates ◽  
Exponential Growth ◽  
Temperature Increase ◽  
Phosphorus Limitation ◽  
Photic Zone ◽  
P Limitation ◽  
P Content ◽  
Phytoplankton Communities ◽  
Coccolithus Pelagicus

Abstract. Rising ocean temperatures will likely increase stratification of the water column and reduce nutrient input into the photic zone. This will increase the likelihood of nutrient limitation in marine microalgae, leading to changes in the abundance and composition of phytoplankton communities, which in turn will affect global biogeochemical cycles. Calcifying algae, such as coccolithophores, influence the carbon cycle by fixing CO2 into particulate organic carbon through photosynthesis (POC production) and into particulate inorganic carbon through calcification (PIC production). As calcification produces a net release of CO2, the ratio of PIC to POC production determines whether coccolithophores act as a source (high PIC / POC) or a sink (low PIC / POC) of atmospheric CO2. We studied the effect of phosphorus (P-) limitation and high temperature on the physiology and the PIC / POC ratio of two subspecies of Coccolithus pelagicus. This large and heavily calcified species is a major contributor to calcite export from the photic zone into deep-sea reservoirs. Phosphorus limitation did not influence exponential growth rates in either subspecies, but P-limited cells had significantly lower cellular P-content. One of the subspecies was subjected to a 5 °C temperature increase from 10 °C to 15 °C, which did not affect exponential growth rates either, but nearly doubled cellular P-content under both high and low phosphate availability. This temperature increase reduced the PIC / POC ratio by 40–60%, whereas the PIC / POC ratio did not differ between P-limited and nutrient-replete cultures when the subspecies were grown near their respective isolation temperature. Both P-limitation and elevated temperature significantly increased coccolith malformations. Our results suggest that a temperature increase may intensify P-limitation due to a higher P-requirement to maintain growth and POC production rates, possibly reducing abundances in a warmer ocean. Under such a scenario C. pelagicus may decrease its calcification rate relative to photosynthesis, thus favouring CO2 sequestration over release. It seems unlikely that P-limitation by itself causes changes in the PIC / POC ratio in this species.

scholarly journals Exponential growth rates in a typed branching diffusion

2007 ◽  
Vol 17 (2) ◽  
pp. 609-653 ◽  
Author(s):  
Y. Git ◽  
J. W. Harris ◽  
S. C. Harris
Keyword(s):  
Growth Rates ◽  
Exponential Growth ◽  
Branching Diffusion
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