The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

In this paper, we propose a worst-case weighted approach to the multi-objective n-person non-zero sum game model where each player has more than one competing objective. Our “worst-case weighted multi-objective game” model supposes that each player has a set of weights to its objectives and wishes to minimize its maximum weighted sum objectives where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto Nash equilibrium concept, which we call “robust-weighted Nash equilibrium”. We prove that the robust-weighted Nash equilibria are guaranteed to exist even when the weight sets are unbounded. For the worst-case weighted multi-objective game with the weight sets of players all given as polytope, we show that a robust-weighted Nash equilibrium can be obtained by solving a mathematical program with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multi-objective game to a supply chain risk management problem under demand uncertainty. By the comparison with the existed weighted approach, we show that our method is more robust and can be more efficiently used for the real-world applications.     

Global stability of partially closed food-chains with resources

We consider the existence and global stability of aq-member equilibrium (1≤q≤n) in partially closed food-chains of lengthn having an abiotic component as resource. We observe that such existence demands bounds of resource supply rate and these bounds are weighted sums of interaction coefficients. Particular results of global sector-stability of partially feasible equilibria of simple food-chains obeying Lotka-Volterra dynamics are shown. Lastly the elasticity of such food-chains when a new species is introduced at the highest trophic level is investigated.     

Fractal dimension of electroencephalographic time series and underlying brain processes

 Fractal dimension has been proposed as a useful measure for the characterization of electrophysiological time series. This paper investigates what the pointwise dimension of electroencephalographic (EEG) time series can reveal about underlying neuronal generators. The following theoretical assumptions concerning brain function were made (i) within the cortex, strongly coupled neural assemblies exist which oscillate at certain frequencies when they are active, (ii) several such assemblies can oscillate at a time, and (iii) activity flow between assemblies is minimal. If these assumptions are made, cortical activity can be considered as the weighted sum of a finite number of oscillations (plus noise). It is shown that the correlation dimension of finite time series generated by multiple oscillators increases monotonically with the number of oscillators. Furthermore, it is shown that a reliable estimate of the pointwise dimension of the raw EEG signal can be calculated from a time series as short as a few seconds. These results indicate that (i) The pointwise dimension of the EEG allows conclusions regarding the number of independently oscillating networks in the cortex, and (ii) a reliable estimate of the pointwise dimension of the EEG is possible on the basis of short raw signals. Received: 1 September 1994/Accepted in revised form: 16 May 1995     

On the adsorption of ions at charged sites in an electric field: II

A study is made of the adsorption of one kind of monovalent positive ion at a long chain of alternating monovalent negative fixed charged (“lattice”) and uncharged (“interstitial”) sites both of one type in an electric field. Considering only nearest neighbor interactions an expression is obtained for the grand partition function. The fractions of sites of both types which are occupied and unoccupied are determined. It is shown that an equilibrium constant can be defined for the adsorption of ions at oppositely charged sites.     

Feedback control for chemostat models

 It is shown that a chemostat with two organisms can be made coexistent by means of feedback control of the dilution rate. Remaining freedom in the feedback law can be used to guarantee robustness or improve particular performance indices. Unfortunately a topological property prevents coexistence by feedback control for chemostats with more than two organisms. We apply our results to control bioreactors aimed at producing commercial products through genetically altered organisms. In all our results the coexistence takes its simplest form: a global asymptotically stable equilibrium point in the interior of the non-negative orthant. Received: 1 November 2001 / Revised version: 19 August 2002 / Published online: 19 November 2002 Mathematics Subject Classification (2000): 34H05, 34D23, 93C10 Keywords or phrases: Chemostat – Feedback control – Coexistence     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

The Robust Weighted Multi-Objective Game

This paper studies a class of multi-objective n-person non-zero sum games through a robust weighted approach where each player has more than one competing objective. This robust weighted multi-objective game model assumes that each player attaches a set of weights to its objectives instead of accessing accurate weights. Each player wishes to minimize its maximum weighted sum objective where the maximization is pointing to the set of weights. To address this new model, a new equilibrium concept-robust weighted Nash equilibrium is obtained. The existence of this new concept is proven on suitable assumptions about the multi-objective payoffs.     

Coexistence of two types on a single resource in discrete time

Armstrong and McGehee (1980) have shown that two species modeled in continuous time can coexist on a single resource provided that one species oscillates autonomously. This paper demonstrates the parallel result in discrete time. I consider a deterministic model of two asexual types in a single patch competing for a single resource, and show that such systems generically produce oscillatory coexistence or bistability if one of the types displays periodic or chaotic behavior in isolation. The conditions for coexistence or bistability are derived in terms of the convexity of the functions describing fitness as a function of resource availability. I also analyze whether or not a stable type, a type with a stable equilibrium population size when considered in isolation, can invade a periodic orbit of an unstable type, and show that the same convexity condition distinguishes these two cases. The widely considered exponential or Ricker model for population dynamics lies on the boundary between the two cases and is highly degenerate in this context.     

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACT

A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.     

A new model for interacting populations—II: Principle of competitive exclusion

The principle of competitive exclusion is investigated within the framework of the solvable model proposed earlier for two-species systems. The results elucidate the recent controversy over the interpretation of experimental data onDrosophila equilibrium. It is shown that the necessary and sufficient conditions for stable coexistence of competing species is that the product of intraspecific rate constants be greater than the product of interspecific rate constants. Inequalities between rate constants for the occurrence of stable equilibriumbelow andabove the line joining single species equilibria are derived. The availability of larger domain of coexistence suggests that the model presented here has the potential to accommodate a wider class of phenomena than the Gause—Volterra model according to which coexistence is possible only above the line of single species equilibrium.     

Local diversity in heterogeneous landscapes: quantitative assessment with a height-structured forest metacommunity model

“Mass effects,” in which “sink populations” of locally inferior competitors are maintained by dispersal from “source populations” elsewhere in the landscape, are thought to play an important role in maintaining plant diversity. However, due to the complexity of most quasi-realistic forest models, there is little theoretical understanding of the strength of mass effects in forests. Here, we develop a metacommunity version of a mathematically and computationally tractable height-structured forest model, the Perfect Plasticity Approximation, to quantify the strength of mass effects (i.e., the degree of mixing of locally dominant and subordinate species) in heterogeneous landscapes comprising different patch types (e.g., soil types). For realistic levels of inter-patch dispersal, mass effects are weak at equilibrium (i.e., in the absence of disturbance), even in some cases where differences in growth, mortality, and fecundity rates between locally dominant and subordinate species are too small to be reliably detected from field data. However, patch-scale transient dynamics are slow following catastrophic disturbance (in which post-disturbance initial abundances are determined exclusively by immigration) so that at any given time, subordinate species are present in appreciable numbers in most patches. Less severe disturbance regimes, in which some seeds or individuals survive the disturbance, should result in faster transient dynamics (i.e., faster approach to the low-diversity equilibrium). Our results suggest that in order for mass effects to play an important role in tree coexistence, niche differences must be strong enough to prevent neutral drift, yet too weak to be reliably detected from field data.     

Some game-theoretical aspects of parasitism and symbiosis

Given a two-person continuous non-constant sum game, a “solution” can be arrived at even in ignorance of the pay-off functions as each player tries to maximize his own pay-off by trial and error. The solution of the game considered here is either the intersection of two “optimal lines” (the case of the stable equilibrium) or parasitism of one player upon the other (the case of unstable equilibrium). The introduction of secondary satisfactions (linear combinations of the pay-offs) affects both the position and the stability of the equilibrium and thus the resulting primary satisfactions (the pay-offs). These effects of the matrix of transformation are investigated. In particular, it is shown that the social optimum, which is also the solution of the game where collusion is allowed, is attained if and only if the columns of the matrix are identical, and that this solution is generally stable. Some suggested connections with biological situations are discussed.     

Inclusive Fitness from Multitype Branching Processes

I use multitype branching processes to study genetic models for the evolution of social behaviour, i.e. behaviours that, when acted out, affect the success of the actor’s neighbours. Here, I suppose an individual bearing a mutant copy of a gene influences the reproductive success of a neighbour by altering its own competitive ability. Approximations based on assumptions about the rareness of the mutant allele and the strength of selection allow me to formulate statements concerning the probability of mutant extinction in terms of inclusive fitness. Inclusive fitness is an idea well known to biologists and can be thought of as a sum of an individual’s fitness and the fitness of each of its relatives, weighted by some measure of genetic relatedness. Previous work has led to some confusion surrounding the definition of the inclusive-fitness effect of a mutant allele when individuals carrying that allele experience demographic conditions that fluctuate randomly. In this paper, I emphasise the link between inclusive fitness and the probability of mutant extinction. I recover standard results for populations of constant size, and I show that inclusive fitness can be used to determine the short-term fate of mutants in the face of stochastic demographic fluctuations. Overall, then, I provide a connection between certain inclusive-fitness-based approaches routinely applied in theoretical studies of social evolution.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210–219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine “interior” and “exterior” species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Numerical and exact solutions for continuum of alleles models

 Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method. Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002 Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection     

带有饱和接触率和垂直传染的两病株病原体的共存性（英文）

讨论了带有饱和接触率和垂直传染的两病株病原体的共存性,获得了共存性存在的充要条件R_2（N_2~＊）〈R_1（N_2~＊）,R_1（N_1~＊）〈R_2（N_1~＊）,并且在共存平衡点存在的条件下证明得到其局部稳定.     

Body size and species coexistence in consumer–resource interactions: A comparison of two alternative theoretical frameworks

Species coexistence involving trophic interactions has been investigated under two theoretical frameworks—partitioning shared resources and accessing exclusive resources. The influence of body size on coexistence is well studied under the exclusive resources framework, but has received less attention under the shared-resources framework. We investigate body-size-dependent allometric extensions of a classical MacArthur-type model where two consumers compete for two shared resources. The equilibrium coexistence criteria are compared against the general predictions of the alternative framework over exclusive resources. From the asymmetry in body size allometry of resource encounter versus demand our model shows, counterintuitively, and contrary to the exclusive resource framework, that a smaller consumer should be competitively superior across a wide range of supplies of the two resource types. Experimental studies are reviewed to resolve this difference among the two frameworks that arise from their respective assumptions over resource distribution. Another prediction is that the smaller consumer may have relatively stronger control over equilibrium resource abundance, and the loss of smaller consumers from a community may induce relatively stronger trophic cascades. Finally, from satiating consumers’ functional response, our model predicts that greater difference among resource sizes can allow a broader range of consumer body sizes to coexist, and this is consistent with the predictions of the alternative framework over exclusive resources. Overall, this analysis provides an objective comparison of the two alternative approaches to understand species coexistence that have heretofore developed in relative isolation. It advances classical consumer–resource theory to show how body size can be an important factor in resource competition and coexistence.     

Changes in the Endogenous Growth Regulators in Bulbous Iris in Bulb-forming and Nonbulb-forming Aspects

Bulblets of 3–4 g of Dutch iris (Iris hollandica) cv.Dominator stored at 20°C and then grown at 15°C developedonly three or four leaves and bulbs formed at the base of eachleaf, whereas when grown at 25°C, they continued to growvegetatively with the development of 10 true leaves and didnot form any new bulb. This demonstrated that bulb formationin bulbous plants can be controlled by environmental factors.The levels of both abscisic acid and auxin activities increasedduring growth under the bulb-forming condition whereas onlyauxin activity increased under the nonbulb-forming condition.The coexistence of both abscisic acid and auxin seems to beessential in the processes of bulb formation. (Received August 22, 1980; Accepted November 26, 1980)