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  • 1
    Electronic Resource
    Electronic Resource
    Global stability in time-delayed single-species dynamics (1986)
    Springer
    Bulletin of mathematical biology 48 (1986), S. 485-492 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Criteria are established for three classes of models of single-species dynamics with a single discrete delay to have a globally asymptotically stable positive equilibrium independent of the length of delay.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462319
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  • 2
    Electronic Resource
    Electronic Resource
    Predator-prey systems with group defence: The paradox of enrichment revisited (1986)
    Springer
    Bulletin of mathematical biology 48 (1986), S. 493-508 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The main concern of this paper is with survival or extinction of predators in models of predator-prey systems exhibiting group defence of the prey. It is shown that if there is no mutual interference among predators, enrichment could result in their extinction. However, if there is mutual interference, the predator population survives (at least deterministically).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462320
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  • 3
    Electronic Resource
    Electronic Resource
    The trade-off between mutual interference and time lags in predator-prey systems (1983)
    Springer
    Bulletin of mathematical biology 45 (1983), S. 991-1004 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We present a Gause predator-prey model incorporating mutual interference among predators, a density-dependent predator death rate and a time lag due to gestation. It is well known that mutual interference is stabilizing, whereas time delays are destabilizing. We show that in combining the two, a long time-lag usually, but not always, destabilizes the system. We also show that increasing delays can cause a bifurcation into periodic solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458826
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  • 4
    Electronic Resource
    Electronic Resource
    Interactions leading to persistence in predator-prey systems with group defence (1988)
    Springer
    Bulletin of mathematical biology 50 (1988), S. 517-530 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract In previous work (Freedman and Wolkowicz, 1986;Bull. math. Biol. 48, 493–508) it was shown that in a predator-prey system where the prey population exhibits group defence, it is possible that enrichment of the environment could lead to extinction of the predator population. In this paper a third population is introduced and criteria are derived under which persistence of all populations will occur. In particular, criteria for a superpredator and for a competitor to stabilize the system in the sense of persistence are analyzed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458850
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  • 5
    Electronic Resource
    Electronic Resource
    Modeling persistence and mutual interference among subpopulations of ecological communities (1985)
    Springer
    Bulletin of mathematical biology 47 (1985), S. 295-304 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model is proposed of an ecological community where some (or all) of the subpopulations exhibit mutual interference. Mutual interference introduces sublinearities which makes the persistence analysis of the community more complex since the model is no longer a dynamical system. A transformation is introduced which yields a dynamical system, thereby making a persistence analysis more tractable. The results are applied to determining top-predator persistence of a simple food chain and to the question of invasibility of a stable community by a new subpopulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460038
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  • 6
    Electronic Resource
    Electronic Resource
    Model of diffusion of oxygen to spheroids grown in stationary medium —I. Complete spherical symmetry (1984)
    Springer
    Bulletin of mathematical biology 46 (1984), S. 205-217 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The use of spheroids as a tumor model has become commonplace since it was discovered that many cell lines can form spheroids when grown on a surface to which the cells cannot attach. This culture system complicates experiments which depend on oxygen supply because the oxygen concentration in the vicinity of a stationary spheroid has not been well defined. We present in this paper solutions to the oxygen diffusion equation for simple geometries: a spheroid in an infinite stationary medium and in a finite spherical stationary medium. Comparison of these solutions provides an estimate of the oxygen supply to a spheroid in a Petri dish. We show that typical spheroids can be expected to cause a substantial depletion of the oxygen in the nearby medium. Any disturbance of the medium or the spheroids will temporarily increase the oxygen supply. We provide a method for estimating the rate of return to equilibrium in the finite cases. These results indicate that the oxygen supply to stationary spheroids can be altered temporarily by small movements or changes in temperature which cause convection currents, or permanently by changes in the depth of the medium.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460069
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  • 7
    Electronic Resource
    Electronic Resource
    Persistence and global stability in a predator-prey model consisting of three prey genotypes with fertility differences (1986)
    Springer
    Bulletin of mathematical biology 48 (1986), S. 469-484 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract A model of a predator-prey interaction, where the prey population consists of three genotypes with random mating and continuous, nonlinear birth and death processes with fertility differences, is proposed. Sufficiency conditions giving the existence of a globally stable equilibrium on one of the coordinate planes are given. This extends results of Freedman and Waltman [J. Math. Biol. 6, 367–374 (1978) andRocky Mountain J. Math. 12, 779–784 (1982)]. In addition, conditions are derived which guarantee the persistence of all components of the populations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02462318
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  • 8
    Electronic Resource
    Electronic Resource
    Persistence in predator-prey systems with ratio-dependent predator influence (1993)
    Springer
    Bulletin of mathematical biology 55 (1993), S. 817-827 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Predator-prey models where one or more terms involve ratios of the predator and prey populations may not be valid mathematically unless it can be shown that solutions with positive initial conditions never get arbitrarily close to the axis in question, i.e. that persistence holds. By means of a transformation of variables, criteria for persistence are derived for two classes of such models, thereby leading to their validity. Although local extinction certainly is a common occurrence in nature, it cannot be modeled by systems which are ratio-dependent near the axes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02460674
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  • 9
    Electronic Resource
    Electronic Resource
    Uniform persistence and flows near a closed positively invariant set (1994)
    Springer
    Journal of dynamics and differential equations 6 (1994), S. 583-600 
    Details
    ISSN: 1572-9222
    Keywords: Positively invariant set ; limit set ; prolongational set ; dissipativeness ; persistence ; 34C35 ; 58F25 ; 92A15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the behavior of a continuous flow in the vicinity of a closed positively invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider varity of ecological models.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02218848
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  • 10
    Electronic Resource
    Electronic Resource
    Further critical cases of the scalar implicit function theorem (1971)
    Springer
    Aequationes mathematicae 7 (1971), S. 121-121 
    Details
    ISSN: 1420-8903
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01818712
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