A sharp threshold for collapse of the random triangular group

Abstract In this paper, a nonlinear nonautonomous model in a rocky intertidal community is studied. The model is composed of two species in a rocky intertidal community and describes a patch occupancy with global dispersal of propagules and occupy each other by individual organisms. Firstly, we study the uniform persistence of the model via differential inequality techniques. Furthermore, a sharp threshold of global asymptotic stability and the existence of a unique almost periodic solution are derived. To prove the main results, we construct an appropriate Lyapunov function whose conditions are easily verified. The assumptions of the model are reasonable, and the results complement previously known ones. An example with specific values of parameters is included for demonstration of theoretical outcomes.

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A multigroup model for cholera dynamics and control

International Journal of Biomathematics ◽

10.1142/s1793524516500017 ◽

2015 ◽

Vol 09 (01) ◽

pp. 1650001 ◽

Cited By ~ 2

Author(s):

Drew Posny ◽

Chairat Modnak ◽

Jin Wang

Keyword(s):

Reproduction Number ◽

Vaccination Strategy ◽

Sharp Threshold ◽

Indirect Transmission ◽

Dynamics And Control ◽

Simulation Results ◽

Transmission Pathways ◽

And Control ◽

Control Study ◽

The Basic Reproduction Number

We propose a general multigroup model for cholera dynamics that involves both direct and indirect transmission pathways and that incorporates spatial heterogeneity. Under biologically feasible conditions, we show that the basic reproduction number R0 remains a sharp threshold for cholera dynamics in multigroup settings. We verify the analysis by numerical simulation results. We also perform an optimal control study to explore optimal vaccination strategy for cholera outbreaks.

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The effects of lowered salinity upon Branchiostoma lanceolatum from the English Channel

Journal of the Marine Biological Association of the United Kingdom ◽

10.1017/s002531540004813x ◽

1981 ◽

Vol 61 (3) ◽

pp. 685-689 ◽

Cited By ~ 1

Author(s):

J. Binyon

Keyword(s):

Body Weight ◽

Osmotic Stress ◽

Sea Water ◽

English Channel ◽

Sharp Threshold ◽

Branchiostoma Lanceolatum

Previous work has indicated that, although in possession of numerous supposedly osmoregulatory structures, Branchiostoma lanceolatum from the English Channel is unable to regulate its body weight in diluted sea water. In vitro measurements of the rate of flagella activity in those organs similarly indicates no increase under hypo-osmotic stress. There is however quite a sharp threshold around 18‰, below which flagella activity ceases quite abruptly. A similar situation obtains with the gill cilia although the animal can remain alive at these salinities for a period of several weeks.

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Topology of random -dimensional cubical complexes

Forum of Mathematics Sigma ◽

10.1017/fms.2021.64 ◽

2021 ◽

Vol 9 ◽

Author(s):

Matthew Kahle ◽

Elliot Paquette ◽

Érika Roldán

Keyword(s):

Fundamental Group ◽

Random Graphs ◽

High Probability ◽

Cubical Complex ◽

Sharp Threshold ◽

Cubical Complexes ◽

Simple Connectivity ◽

Dimensional Cube ◽

Dimensional Face ◽

Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

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The Total Acquisition Number of Random Graphs

The Electronic Journal of Combinatorics ◽

10.37236/5327 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 1

Author(s):

Deepak Bal ◽

Patrick Bennett ◽

Andrzej Dudek ◽

Paweł Prałat

Keyword(s):

Random Graph ◽

Random Graphs ◽

High Probability ◽

Sharp Threshold ◽

Positive Weight ◽

Minimum Value ◽

Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

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Sharp Threshold Functions for Random Intersection Graphs via a Coupling Method

The Electronic Journal of Combinatorics ◽

10.37236/523 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 30

Author(s):

Katarzyna Rybarczyk

Keyword(s):

Perfect Matching ◽

Hamilton Cycle ◽

Coupling Method ◽

New Method ◽

Intersection Graphs ◽

Sharp Threshold ◽

Threshold Functions

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for $k$–connectivity, perfect matching containment and Hamilton cycle containment.

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Random high-dimensional orders

Advances in Applied Probability ◽

10.1017/s0001867800046292 ◽

1995 ◽

Vol 27 (01) ◽

pp. 161-184 ◽

Cited By ~ 1

Author(s):

Béla Bollobás ◽

Graham Brightwell

Keyword(s):

Partial Order ◽

Rapid Growth ◽

Entire Range ◽

Maximum Degree ◽

Threshold Function ◽

High Dimensional ◽

Sharp Threshold ◽

Threshold Functions

The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1] k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n. We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n). We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k. We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).

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