Matching Items (13)

Application of Individual Based Models in Modeding Brain Tumor

Description

Cancer modeling has brought a lot of attention in recent years. It had been proven to be a difficult task to model the behavior of cancer cells, since little about

Cancer modeling has brought a lot of attention in recent years. It had been proven to be a difficult task to model the behavior of cancer cells, since little about the "rules" a cell follows has been known. Existing models for cancer cells can be generalized into two categories: macroscopic models which studies the tumor structure as a whole, and microscopic models which focus on the behavior of individual cells. Both modeling strategies strive the same goal of creating a model that can be validated with experimental data, and is reliable for predicting tumor growth. In order to achieve this goal, models must be developed based on certain rules that tumor structures follow. This paper will introduce how such rules can be implemented in a mathematical model, with the example of individual based modeling.

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Agent

Created

Date Created
  • 2016-12

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Jump Dynamics

Description

There are multiple mathematical models for alignment of individuals moving within a group. In a first class of models, individuals tend to relax their velocity toward the average velocity of

There are multiple mathematical models for alignment of individuals moving within a group. In a first class of models, individuals tend to relax their velocity toward the average velocity of other nearby neighbors. These types of models are motivated by the flocking behavior exhibited by birds. Another class of models have been introduced to describe rapid changes of individual velocity, referred to as jump, which better describes behavior of smaller agents (e.g. locusts, ants). In the second class of model, individuals will randomly choose to align with another nearby individual, matching velocities. There are several open questions concerning these two type of behavior: which behavior is the most efficient to create a flock (i.e. to converge toward the same velocity)? Will flocking still emerge when the number of individuals approach infinity? Analysis of these models show that, in the homogeneous case where all individuals are capable of interacting with each other, the variance of the velocities in both the jump model and the relaxation model decays to 0 exponentially for any nonzero number of individuals. This implies the individuals in the system converge to an absorbing state where all individuals share the same velocity, therefore individuals converge to a flock even as the number of individuals approach infinity. Further analysis focused on the case where interactions between individuals were determined by an adjacency matrix. The second eigenvalues of the Laplacian of this adjacency matrix (denoted ƛ2) provided a lower bound on the rate of decay of the variance. When ƛ2 is nonzero, the system is said to converge to a flock almost surely. Furthermore, when the adjacency matrix is generated by a random graph, such that connections between individuals are formed with probability p (where 01/N. ƛ2 is a good estimator of the rate of convergence of the system, in comparison to the value of p used to generate the adjacency matrix..

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Agent

Created

Date Created
  • 2018-05

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Incorporating Space and Stochasticity in an Evolutionary Game Theory Model

Description

This thesis explores and explains a stochastic model in Evolutionary Game Theory introduced by Dr. Nicolas Lanchier. The model is a continuous-time Markov chain that maps the two-dimensional lattice into

This thesis explores and explains a stochastic model in Evolutionary Game Theory introduced by Dr. Nicolas Lanchier. The model is a continuous-time Markov chain that maps the two-dimensional lattice into the strategy space {1,2}. At every vertex in the grid there is exactly one player whose payoff is determined by its strategy and the strategies of its neighbors. Update times are exponential random variables with parameters equal to the absolute value of the respective cells' payoffs. The model is connected to an ordinary differential equation known as the replicator equation. This differential equation is analyzed to find its fixed points and stability. Then, by simulating the model using Java code and observing the change in dynamics which result from varying the parameters of the payoff matrix, the stochastic model's phase diagram is compared to the replicator equation's phase diagram to see what effect local interactions and stochastic update times have on the evolutionary stability of strategies. It is revealed that in the stochastic model altruistic strategies can be evolutionarily stable, and selfish strategies are only evolutionarily stable if they are more selfish than their opposing strategy. This contrasts with the replicator equation where selfishness is always evolutionarily stable and altruism never is.

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Agent

Created

Date Created
  • 2013-12

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Heterophilious Dynamics Enhances Consensus

Description

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents,” with the tendency

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents,” with the tendency to adjust to their “environmental averages.” This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents, and what types of “rules of engagement” influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish, or cells, rendezvous of mobile agents, and, in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological, and physical sciences assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, “similarity breeds connection” reflects our intuition that increasing the intensity of alignment as the difference of positions decreases is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily, the tendency to bond more with those who are different rather than with those who are similar, plays a decisive role in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.

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Agent

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Date Created
  • 2013-11-30

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Clustering and asymptotic behavior in opinion formation

Description

We investigate the long time behavior of models of opinion formation. We consider the case of compactly supported interactions between agents which are also non-symmetric, including for instance the so-called

We investigate the long time behavior of models of opinion formation. We consider the case of compactly supported interactions between agents which are also non-symmetric, including for instance the so-called Krause model. Because of the finite range of interaction, convergence to a unique consensus is not expected in general. We are nevertheless able to prove the convergence to a final equilibrium state composed of possibly several local consensus. This result had so far only been conjectured through numerical evidence. Because of the non-symmetry in the model, the analysis is delicate and is performed in two steps: First using entropy estimates to prove the formation of stable clusters and then studying the evolution in each cluster. We study both discrete and continuous in time models and give rates of convergence when those are available.

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Created

Date Created
  • 2014-12-01

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Optimal sampling for linear function approximation and high-order finite difference methods over complex regions

Description

I focus on algorithms that generate good sampling points for function approximation. In 1D, it is well known that polynomial interpolation using equispaced points is unstable. On the other hand,

I focus on algorithms that generate good sampling points for function approximation. In 1D, it is well known that polynomial interpolation using equispaced points is unstable. On the other hand, using Chebyshev nodes provides both stable and highly accurate points for polynomial interpolation. In higher dimensional complex regions, optimal sampling points are not known explicitly. This work presents robust algorithms that find good sampling points in complex regions for polynomial interpolation, least-squares, and radial basis function (RBF) methods. The quality of these nodes is measured using the Lebesgue constant. I will also consider optimal sampling for constrained optimization, used to solve PDEs, where boundary conditions must be imposed. Furthermore, I extend the scope of the problem to include finding near-optimal sampling points for high-order finite difference methods. These high-order finite difference methods can be implemented using either piecewise polynomials or RBFs.

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Agent

Created

Date Created
  • 2019

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Swarming in bounded domains

Description

Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion

Swarms of animals, fish, birds, locusts etc. are a common occurrence but their coherence and method of organization poses a major question for mathematics and biology.The Vicsek and the Attraction-Repulsion are two models that have been proposed to explain the emergence of collective motion. A major issue for the Vicsek Model is that its particles are not attracted to each other, leaving the swarm with alignment in velocity but without spatial coherence. Restricting the particles to a bounded domain generates global spatial coherence of swarms while maintaining velocity alignment. While individual particles are specularly reflected at the boundary, the swarm as a whole is not. As a result, new dynamical swarming solutions are found.

The Attraction-Repulsion Model set with a long-range attraction and short-range repulsion interaction potential typically stabilizes to a well-studied flock steady state solution. The particles for a flock remain spatially coherent but have no spatial bound and explore all space. A bounded domain with specularly reflecting walls traps the particles within a specific region. A fundamental refraction law for a swarm impacting on a planar boundary is derived. The swarm reflection varies from specular for a swarm dominated by

kinetic energy to inelastic for a swarm dominated by potential energy. Inelastic collisions lead to alignment with the wall and to damped pulsating oscillations of the swarm. The fundamental refraction law provides a one-dimensional iterative map that allows for a prediction and analysis of the trajectory of the center of mass of a flock in a channel and a square domain.

The extension of the wall collisions to a scattering experiment is conducted by setting two identical flocks to collide. The two particle dynamics is studied analytically and shows a transition from scattering: diverging flocks to bound states in the form of oscillations or parallel motions. Numerical studies of collisions of flocks show the same transition where the bound states become either a single translating flock or a rotating (mill).

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Agent

Created

Date Created
  • 2015

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The economics of need-based transfers

Description

Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African

Need-based transfers (NBTs) are a form of risk-pooling in which binary welfare exchanges

occur to preserve the viable participation of individuals in an economy, e.g. reciprocal gifting

of cattle among East African herders or food sharing among vampire bats. With the

broad goal of better understanding the mathematics of such binary welfare and risk pooling,

agent-based simulations are conducted to explore socially optimal transfer policies

and sharing network structures, kinetic exchange models that utilize tools from the kinetic

theory of gas dynamics are utilized to characterize the wealth distribution of an NBT economy,

and a variant of repeated prisoner’s dilemma is analyzed to determine whether and

why individuals would participate in such a system of reciprocal altruism.

From agent-based simulation and kinetic exchange models, it is found that regressive

NBT wealth redistribution acts as a cutting stock optimization heuristic that most efficiently

matches deficits to surpluses to improve short-term survival; however, progressive

redistribution leads to a wealth distribution that is more stable in volatile environments and

therefore is optimal for long-term survival. Homogeneous sharing networks with low variance

in degree are found to be ideal for maintaining community viability as the burden and

benefit of NBTs is equally shared. Also, phrasing NBTs as a survivor’s dilemma reveals

parameter regions where the repeated game becomes equivalent to a stag hunt or harmony

game, and thus where cooperation is evolutionarily stable.

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Agent

Created

Date Created
  • 2018

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Modeling collective motion of complex systems using agent-based models & macroscopic models

Description

The main objective of mathematical modeling is to connect mathematics with other scientific fields. Developing predictable models help to understand the behavior of biological systems. By testing models, one can

The main objective of mathematical modeling is to connect mathematics with other scientific fields. Developing predictable models help to understand the behavior of biological systems. By testing models, one can relate mathematics and real-world experiments. To validate predictions numerically, one has to compare them with experimental data sets. Mathematical modeling can be split into two groups: microscopic and macroscopic models. Microscopic models described the motion of so-called agents (e.g. cells, ants) that interact with their surrounding neighbors. The interactions among these agents form at a large scale some special structures such as flocking and swarming. One of the key questions is to relate the particular interactions among agents with the overall emerging structures. Macroscopic models are precisely designed to describe the evolution of such large structures. They are usually given as partial differential equations describing the time evolution of a density distribution (instead of tracking each individual agent). For instance, reaction-diffusion equations are used to model glioma cells and are being used to predict tumor growth. This dissertation aims at developing such a framework to better understand the complex behavior of foraging ants and glioma cells.

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Created

Date Created
  • 2019

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Weak measure-valued solutions to a nonlinear conservation law modeling a highly re-entrant manufacturing system

Description

The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and

The main part of this work establishes existence, uniqueness and regularity properties of measure-valued solutions of a nonlinear hyperbolic conservation law with non-local velocities. Major challenges stem from in- and out-fluxes containing nonzero pure-point parts which cause discontinuities of the velocities. This part is preceded, and motivated, by an extended study which proves that an associated optimal control problem has no optimal $L^1$-solutions that are supported on short time intervals.

The hyperbolic conservation law considered here is a well-established model for a highly re-entrant semiconductor manufacturing system. Prior work established well-posedness for $L^1$-controls and states, and existence of optimal solutions for $L^2$-controls, states, and control objectives. The results on measure-valued solutions presented here reduce to the existing literature in the case of initial state and in-flux being absolutely continuous measures. The surprising well-posedness (in the face of measures containing nonzero pure-point part and discontinuous velocities) is directly related to characteristic features of the model that capture the highly re-entrant nature of the semiconductor manufacturing system.

More specifically, the optimal control problem is to minimize an $L^1$-functional that measures the mismatch between actual and desired accumulated out-flux. The focus is on the transition between equilibria with eventually zero backlog. In the case of a step up to a larger equilibrium, the in-flux not only needs to increase to match the higher desired out-flux, but also needs to increase the mass in the factory and to make up for the backlog caused by an inverse response of the system. The optimality results obtained confirm the heuristic inference that the optimal solution should be an impulsive in-flux, but this is no longer in the space of $L^1$-controls.

The need for impulsive controls motivates the change of the setting from $L^1$-controls and states to controls and states that are Borel measures. The key strategy is to temporarily abandon the Eulerian point of view and first construct Lagrangian solutions. The final section proposes a notion of weak measure-valued solutions and proves existence and uniqueness of such.

In the case of the in-flux containing nonzero pure-point part, the weak solution cannot depend continuously on the time with respect to any norm. However, using semi-norms that are related to the flat norm, a weaker form of continuity of solutions with respect to time is proven. It is conjectured that also a similar weak continuous dependence on initial data holds with respect to a variant of the flat norm.

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Agent

Created

Date Created
  • 2019