American Institute of Mathematical Sciences

Advanced
July  2010, 14(1): 17-40. doi: 10.3934/dcdsb.2010.14.17

Community resilience in collaborative learning

Nicolás M. Crisosto 1, , Christopher M. Kribs-Zaleta 2, , Carlos Castillo-Chávez 3, and  Stephen Wirkus 4,

1. 

California State University, Channel Islands, One University Drive, Camarillo CA 93012, United States
2. 

Departments of Mathematics and Curriculum & Instruction, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States
3. 

Mathematical, Computational Modeling Sciences Center, PO Box 871904, Arizona State University, Tempe, AZ 85287
4. 

Mathematical & Natural Sciences Division, Arizona State University, Mail Code 2352, P.O. Box 37100, Phoenix, AZ 85069-7100, United States

Received  July 2009 Revised  December 2009 Published  April 2010

This paper introduces a simplified dynamical systems framework for the study of the mechanisms behind the growth of cooperative learning in large communities. We begin from the simplifying assumption that individual-based learning focuses on increasing the individual's "fitness" while collaborative learning may result in the increase of the group's fitness. It is not the objective of this paper to decide which form of learning is more effective but rather to identify what types of social communities of learners can be constructed via collaborative learning. The potential value of our simplified framework is inspired by the tension observed between the theories of intellectual development (individual to collective or vice versa) identified with the views of Piaget and Vygotsky. Here they are mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. The community is generated from sequences of successful "contacts'' between various types of individuals, which generate multiple nonlinearities in the corresponding differential equations that form the model. A bifurcation analysis of the model provides an explanation for the impact of individual learning on community intellectual development, and for the resilience of communities constructed via multilevel epidemiological contact processes, which can survive even under conditions that would not allow them to arise. This simple cooperative framework thus addresses the generalized belief that sharp community thresholds characterize separate learning cultures. Finally, we provide an example of an application of the model. The example is autobiographical as we are members of the population in this "experiment".
Keywords: cooperative learning, backward bifurcation., Population biology.
Mathematics Subject Classification: Primary: 92D25, 34C23; Secondary: 91E9.
Citation: Nicolás M. Crisosto, Christopher M. Kribs-Zaleta, Carlos Castillo-Chávez, Stephen Wirkus. Community resilience in collaborative learning. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 17-40. doi: 10.3934/dcdsb.2010.14.17
[1]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
[2]

Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895
[3]

Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495
[4]

Lianzhang Bao, Zhengfang Zhou. Traveling wave in backward and forward parabolic equations from population dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1507-1522. doi: 10.3934/dcdsb.2014.19.1507
[5]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
[6]

Sumei Li, Yicang Zhou. Backward bifurcation of an HTLV-I model with immune response. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 863-881. doi: 10.3934/dcdsb.2016.21.863
[7]

Muntaser Safan, Klaus Dietz. On the eradicability of infections with partially protective vaccination in models with backward bifurcation. Mathematical Biosciences & Engineering, 2009, 6 (2) : 395-407. doi: 10.3934/mbe.2009.6.395
[8]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[9]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[10]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[11]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[12]

Liming Cai, Jicai Huang, Xinyu Song, Yuyue Zhang. Bifurcation analysis of a mosquito population model for proportional releasing sterile mosquitoes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6279-6295. doi: 10.3934/dcdsb.2019139
[13]

Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749
[14]

Benjamin H. Singer, Denise E. Kirschner. Influence of backward bifurcation on interpretation of $R_0$ in a model of epidemic tuberculosis with reinfection. Mathematical Biosciences & Engineering, 2004, 1 (1) : 81-93. doi: 10.3934/mbe.2004.1.81
[15]

Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515
[16]

Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i
[17]

Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[18]

Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041
[19]

Avner Friedman. PDE problems arising in mathematical biology. Networks & Heterogeneous Media, 2012, 7 (4) : 691-703. doi: 10.3934/nhm.2012.7.691
[20]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences