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1

Effect of dispersal in single-species discrete diffusion systems with source-sink patches

Elbetch Bilel

Mathematica Applicanda

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2023

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Vol. 51, no. 1

51--97

EN

A multi-patch source-sink model with and without intraspecific competition in the sink patches is considered. First, we study the dynamics of the model when the matrix of migration is irreducible and reducible. We show that, there is a threshold number of source patches such that the population potentially becomes extinct below the threshold and established above the threshold. Next, used the theory of singular perturbation and theorem of Tikhonov, in the case of perfect mixing, i.e. when the diffusion rate tends to infinity, we calculate the equilibrium of the model and we give a good approximations of the solutions in this case. Second, we determine, in some particular cases, the conditions under which fragmentation and the existence of sink patches can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities of the source patches. Finally, we study the effect of the rapid growth of the population in source patches and the rapid death of the population in sink patches on the dynamics of the total equilibrium population and on the coexistence of the populations in the patches.

PL

Rozważany jest model wielogatunkowy enklaw wykluczanych ekologicznie z i bez konkurencji wewnątrzgatunkowej w enklawach zanikających. Najpierw badamy dynamikę modelu, gdy macierz migracji jest nieredukowalna i edukowalna. Pokazujemy, że istnieje progowa liczba enklaw, która powoduje, że populacja potencjalnie wymiera poniżej progu i jest ustablilizowana powyżej pewnego pozomu. Następnie, korzystając z teorii perturbacji pojedynczych i twierdzenia Tichonowa, w przypadku doskonałego mieszania, tj. gdy współczynnik dyfuzji dąży do nieskończoności, obliczamy równowagę modelu i podajemy dobre przybliżenia rozwiązań w tym przypadku. Po drugie, określamy, w pewnych szczególnych przypadkach, warunki, w których fragmentacja i istnienie enklaw może prowadzić do tego, że całkowita populacja w równowadze jest większa lub mniejsza niż suma populacji enklaw. Wreszcie, badamy wpływ szybkiego wzrostu populacji źródłowej i szybkiego wymierania populacji populacji zlewowej na dynamikę całkowitej populacji równowagi i na koegzystencję gatunków.

2

The Role of Microhabitat and Water Level in Regulating the Small-Scale Distribution, Seasonal Abundance and Overwintering Success of the Protected Snail Vertigo moulinsiana in a Natural Wetland

Lipińska Anna M., Ćmiel Adam M., Kwaśna Dorota, Myzyk Stanisław, Zając Katarzyna, Zając Tadeusz

Polish Journal of Ecology

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2020

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Vol. 68, nr 3

229--241

EN

Despite the high taxonomic diversity, wide distribution and important ecosystem functions of snails, their ecology remains obscure, and this can hamper their conservation. Overwintering in the studied population of Vertigo moulinsiana is effective enough to allow the majority of it to survive. The population dynamics of V. moulinsiana is specific, where the summer boom in a suboptimal habitat covering a small area (wet patches of sweet grass) can determine the snail's range over much larger areas. Our results suggest that humidity could have been the major factor governing population increase. The summer boom is caused by juveniles appearing in very large numbers during this period, thus the peak height is mostly influenced by the considerable susceptibility of juveniles and eggs to desiccation. The limiting influence of humidity is evident in the unequal spatial distribution of individuals: the wetter the substrate, the greater the abundance of snails. As a consequence, the boom may be spatially regulated and only occurs in wet habitats, even though the snail actually occupies a much larger area. Our results indicate that this V. moulinsiana population does not follow typical models constructed for vertebrates or other animals with high dispersal abilities, like the Ideal Free Distribution (IFD) model, and requires a specific approach. Its specific spatiotemporal pattern should be taken into account during monitoring surveys and conservation actions.

3

Revisiting the Logistic Growth with Random Disturbances

Klimasara Paweł

Mathematica Applicanda

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2019

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Vol. 47, no. 2

177--186

EN

We reconsider a one-dimensional probabilistic model of a fire-induced tree-grass coexistence in savannas introduced by D'Odorico, Laio and Ridol in [5]. We rewrite it as a logistic growth model with random tree biomass losses caused by fire occurring at random times. We study it by using the stochastic semigroup theory and we give new suffcient conditions for the existence and stability of a unique stationary density of woody biomass.

PL

Modele populacyjne oparte o równanie logistyczne wciąż są popularne w modelowaniu ekosytemów i pozwalają lepiej zrozumieć różne zjawiska. W tym artykule rozważamy prosty 1-wymiarowy model sawanny zaproponowany przez D'Odorico, Laio i Ridolfi'ego w pracy [5], który jest modelem współistnienia traw i drzew na sawannach indukowanego losowymi pożarami. Jednak zamiast wprowadzać ubytki biomasy spowodowane występowaniem pożarów bezpośrednio do równań modelu, definiujemy odpowiedni proces stochastyczny. Następnie badamy go z wykorzystaniem teorii półgrup stochastycznych. Zasadniczym wynikiem jest twierdzenie 3.1 określające, kiedy przedstawiony model może opisywać stabilne współistnienie traw i drzew charakterystyczne dla sawann. Mianowicie przy spełnionym warunku (7) istnieje jedyny absolutnie ciągły rozkład stacjonarny biomasy drzew, do którego cały układ będzie dążył, natomiast w sytuacji (8) taki rozkład nie istnieje. Powyższy wynik można łatwo przenieść na wyższe wymiary i zastosować np. w dwuwymiarowym modelu podanym w poprzedniej pracy (na ten temat) autora i Marty Tyran-Kamińskiej [7].

4

Parameter identification and estimation for stage-structured population models

Coll Carmen, Sánchez Elena

International Journal of Applied Mathematics and Computer Science

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2019

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Vol. 29, no. 2

327--336

EN

A stage-structured population model with unknown parameters is considered. Our purpose is to study the identifiability of the model and to develop a parameter estimation procedure. First, we analyze whether the parameter vector can or cannot uniquely be determined with the knowledge of the input-output behavior of the model. Second, we analyze how the information in the experimental data is translated into the parameters of the model. Furthermore, we propose a process to improve the recursive values of the parameters when successive observation data are considered. The structure of the state matrix leads to an analysis of the inverse of a sum of rank-one matrices.

5

Spatiotemporal variation of drift and upstream movements of the amphipod Gammarus fossarum in a small unaltered stream

Žganec K., Gottsein S., Hudina S.

Polish Journal of Ecology

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2013

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Vol. 61, nr 4

769--784

EN

Stream invertebrates use ‘drift’ or downstream transport in the water column to disperse, but they also actively move upstream against current. These movements have important role in re-colonization of empty stream reaches and in regulation of population size. For more than half a century, research of stream drift was stimulated by the problem of ‘drift paradox’, i.e. the question how upstream reaches of streams remain inhabited despite constant and often ample losses of individuals by drift. In this study we examined drift and upstream movements of the amphipod Gammarus fossarum through monthly sampling over one-year period at two sites in the upper and middle course of a small unaltered stream in the northwest Croatia. Differences in tendency to drift and to move upstream and effects of environmental factors (temperature, flow, population density and detritus dry weight) on the movement dynamics on both temporal and spatial scale were examined for three size categories (adults, subadults and juveniles). While seasonal variations of drift and upstream movements were similar at two examined sites, at the middle course we recorded higher density, and higher drift and upstream movement rates. In both drift and upstream movements, adults were the most represented size category. In subadults and juveniles, both type of movements depended on population density. Adult upstream movements peaked in November and May/June, with massive upstream movements in May in the middle course. Seasonality of diurnal and nocturnal drift was very similar at both sites, while seasonality of diurnal and nocturnal upstream movements differed between the sites. Out of four examined independent environmental factors, density and temperature had the most important positive effect on spatiotemporal variations of diurnal drift, while nocturnal drift was more strongly affected by density and flow. Temperature was the most significant driver of diurnal upstream movements, while nocturnal upstream movements were most significantly positively affected by density. Massive upstream movements were driven by overcrowding and enabled adults to colonize microhabitats where food is more abundant and competition is less severe.

6

Formation and dynamics of the metapopulation of Senecio rivularis (Waldst. & Kit.) DC. (Asteraceae) at the limit of its geographical range : where, when, by what means?

Czarnecka B.

Polish Journal of Ecology

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2011

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Vol. 59, nr 2

263-278

EN

In the studies on the metapopulation system two aspects are important: cognitive and practical. The former indicates that a metapopulation is a demographic structure of a higher level than coenotic or local populations and aims at recognition of relations between particular populations of a given species. The practical aspect reflects the issues of the dynamics and/or the extinction rate in particular populations and indicates potential habitats which can be colonized. The study of the Senecio rivularis metapopulation is created by relatively abundant population of ragwort in the Roztocze Highlands, Roztocze National Park, south-eastern Poland, recognized as the source population and three other called marginal or sink populations as well as potentially suitable habitats which are distributed as islands in adjacent hydrogenic areas. The distances between particular populations range from 1.3 to 6.5 km. As a long-lived, iteroparous rhizomatous clonal plant, insect-pollinated and anemoc anemochoric one, so having adaptations both for 'escape in time' and 'escape in space', S. rivularis may be treated as the model object of the study on the metapopulation level. The following hypotheses were put forward: (1) the spatial pattern, size and life-stage structures of particular populations reflect the time of their origin in the meta-system; (2) hydrochory is a complementary or alternative way of migration of anemochoric diaspores; (3) local watercourses and hydro-technical activity affect the formation and dynamics of S. rivularis metapopulation. The multi-aspect studies of the source population were conducted in the years 1987.2007. They aimed to establish the density, spatial pattern and life-stage structure of the S. rivularis population, both at a large scale, i.e. in the whole population (three times) and at a small scale, i.e. on the level of three particular clumps (four times). The size structure of generative genets and ramets in the whole population was aleborated five times. In the greatest of the other populations in the meta-system, the size structure was elaborated twice (2002, 2007) while the study of spatial and life-stage structures was done only once, in the first period, because of the disturbance of a greater part of the population in the next years. In the two remaining localities all plants were counted thanks to small abundance of the populations. Considerable changes in the spatial organization and life-stage structure of genets in the source and sink populations exhibited that colonization in the meta-system of S. rivularis proceeds in accordance with an island model of the 'stepping stone' type, downstream the river. Renaturalization of water relations of river catchment areas and construction of the so-called small retention objects affect the formation and dynamics of S. rivularis metapopulation which reflects at least some dynamics phases observed at the level of the source population: progressive, stable, fluctuation and/or regressive, and extinction phases. Knowledge of the density, life-stage and size structures of plants in a population facilitates a highly reliable determination of the development phase of a given local population in the meta-system and prognosis of its future.

7

Effect of light competition with filamentous algae on the population dynamics and development of the moss species Warnstorfia exannulata in a softwater lake

Szmeja J., Bociąg K., Merdalski M.

Polish Journal of Ecology

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2010

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Vol. 58, nr 2

221-230

EN

The effect of a massive bloom of filamentous algae on the long-term abundance dynamics of the moss Warnstorfia exannulata (B., S. & G.) Loeske was studied in an acidic lowproductivity lake in NW Poland. Individuals were counted on 4 experimental plots, 1 x 1 m each, at a depth of 2.5 m. The studies were conducted for 36 months, every 30 days, by SCUBA diving. Over the three years the seasonal changes in water pH, conductivity, HCO3 - concentration, sediment hydration and pH were not statistically significant (P> 0.05), and light intensity was higher in winter than in summer. Over the three years 4 consecutive stages of population development were observed: regeneration, stabilisation of abundance (fluctuations), regression and repeated regeneration. These stages appeared and lasted at different temperatures and light intensity. The greatest monthly increase in abundance took place in summer and lasted until the beginning of autumn under conditions of high temperature and relatively low light intensity. Long-term abundance variations were strongly correlated with the appearance of filamentous algae, less strongly with the dynamics of the dominant species (Sphagnum denticulatum), and the least with water temperature, light intensity, water and sediment pH, conductivity and HCO3 - concentration. As a result of a massive bloom of these algae, light intensity decreased in the water, Warnstorfia shed its leaves, and then the entire population disappeared. In winter that year it began to regenerate from vegetative propagules (leafless shoots), which were the remains of individuals growing before the filamentous algae appeared. In softwater lakes such massive algal blooms are a common phenomenon which limits light transmission to the substrate and as a result leads to changes in the submerged vegetation structure, especially that of bryophytes.

8

Flowering and generative reproduction in isolated populations of endangered species Carlina onopordifolia Bessr (Asteraceae) in Poland

Denisiuk Z., Chmura D., Adamski P.

Polish Journal of Ecology

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2009

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Vol. 57, nr 1

89-97

EN

Carlina onopordifolia is longlived monocarpic perennial plant, which reproduces only generatively, however, it flowers and fruits only once in lifespan (10-20 yrs) and after seeds development it dies. Due to this type of reproduction the abundance of population depends on amount of produced seeds, moreover the plant flowers more abundantly every 2-3 years. The plant is highly specialized in respect to habitat and micro-climate conditions. It grows on calcareous and sun-heated slopes, where xerothermic grasslands grow with contribution of species typical for steppe vegetation. In Poland it is a very rare species known only from seven isolated natural stands in south and east part of the country. Differences and year-to-year fluctuations in the frequency and the abundance of generative and vegetative specimens in natural stands were studied in 2006 and 2007 years. The regional differences in leaf rosette size and size of inflorescences were examined. The abundance of population in all stands increased from ca. 13600 plants in 2006 to ca. 31950 in 2007. The G test revealed significant decrease in frequency of flowering plants vs. barren plants between the years in four of all seven stands. The significant differences in mean diameter of leaf rosettes (45; 49; 52 cm) and mean diameter of anthodium of generative specimens (7.17; 7.74; 8.27 cm) were observed in three the most geographically distinct and the most abundant populations. The mean values of leave rosette (44.20; 49.31 cm) and sizes of anthodiums (6.05; 7.61 cm) also changed significantly between the years 2006 and 2007, respectively for all data. It can be concluded that in all stands of the thistle in Poland the synchronization of blooming phase takes place. The observations implicate conclusions important for practice of monitoring which must be conducted at least for two vegetative seasons covering the whole phenology of the species in order to prevent the artifacts.

9

Parthenogenesis and gamogenesis in seasonal succession of chydorids (Crustacea, Chydoridae) in three low-productive lakes as observed with activity traps

Nevalainen L.

Polish Journal of Ecology

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2008

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Vol. 56, nr 1

85-97

EN

The aim of the present study was to determine the variance in abundance of parthenogenetic (asexual) and gamogenetic (sexual) individuals among chydorids (Crustacea, Anomopoda, Chydoridae). The chydorids were monitored quantitatively with stationary activity traps in three lakes in southern Finland at 2-week intervals during the open-water season (early May-late October) in 2006. The lakes chosen for the study were low-productive forest lakes; Lake Kalatoin was dystrophic fish-free lake and lakes Tuhkuri and Iso Lehmalampi were oligotrophic. The abundance of trapped individuals varied widely among the lakes and during the sampling period presumably due to site-specific environmental conditions (available microhabitats, food, shelter). The abundance was highest in the dystrophic Lake Kalatoin (max. 43 x 10[^3] m[^2] trapday[^-1] in June) and clearly lower in the two oligotrophic lakes (max. 8.5 x 10[^3] m[^-2] trapday[^-1] in Lake Tuhkuri in mid-July and max. 2.2 x 10[^3] m[^-2] trapday[^1] in Lake Iso Lehmalampi in mid-October). The chydorids exhibited unique sexual reproduction patterns among sampling sites and populations as the abundance of trapped gamogenetic individuals differed, suggesting habitat- and population-specific patterns in gamogenesis. In lakes Kalatoin and Iso Lehmalampi gamogenetic individuals were caught in the traps during the autumn with maxima of 2.2 and 1.6 x 10[^3] m[^-2] trapday[^-1], respectively, but in Lake Tuhkuri no gamogenetic individuals were encountered during the autumn. Although Alonella nana (Baird) was most abundantly trapped species in all the lakes, its gamogenetic individuals were trapped numerously only in Lake Kalatoin (max. ca. 1.5 x l0[^3] m[^-2] trapday[^-1]). The results suggest that gamogenesis in chydorids is a more complex phenomenon than previously believed.

10

Home ranges, nest sites and population dynamics ofthe forest dormouse Dryomys nitedula (Pallas)in an oak-hornbeam forest: a live-trappingand radio-tracking study

Ściński Michał, Borowski Zbigniew

Polish Journal of Ecology

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2006

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Vol. 54, nr 3

391-396

EN

Population dynamics, space use and interspecific interactions of the forest dormouse Dryomys nitedula were studied for four years in an oak-hornbeam forest in the Białowieża Primeval Forest (NE Poland). Animals were captured with live-traps and marked by ear-tattooing. Population density ranged from 1.4 to 18.6 individuals x 10 ha^-1. Dormice reproduced in all years of study (2001-2004). Offspring were born in June and first juveniles were caught at the beginning of August. The last dormice were captured at the end of September. Three forest dormice (1 male and 2 females) were radio-tracked during first half of June 2001. Male home range area was larger (4.2 ha) than home range areas of females (0.75 and 0.73 ha). The longest distance between two consecutive daytime nest sites was 275 m for the male and 126 m for the females. Average home range length calculated from radio-tracked individuals (218 m) was larger than the one from live-trapped dormice (93 m). Dormice used bird nest boxes and tree hollows for daytime nest sites. They preferred nesting in the nest boxes previously occupied by birds and they used old bird nest material for the nest construction. Home ranges of radio-tracked forest dormice overlapped widely with home ranges of fat dormice Glis glis. Most of live-traps used by the forest dormouse were also visited by the fat dormouse.

11

Asymptotic behaviour of a discrete dynamical system generated by a simple evolutionary process

Karcz-Dulęba I.

International Journal of Applied Mathematics and Computer Science

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2004

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Vol. 14, no 1

79-90

EN

A simple model of phenotypic evolution is introduced and analysed in a space of population states. The expected values of the population states generate a discrete dynamical system. The asymptotic behaviour of the system is studied with the use of classical tools of dynamical systems. The number, location and stability of fixed points of the system depend on parameters of a fitness function and the parameters of the evolutionary process itself. The influence of evolutionary process parameters on the stability of the fixed points is discussed. For large values of the standard deviation of mutation, fixed points become unstable and periodical orbits arise. An analysis of the periodical orbits is presented.

12

Solvability and Asymptotic Behavior of a Population Problem Taking into Account Random Mating and Females' Pregnancy

Skakauskas V.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

37-61

EN

Two deterministic age-sex-structured population dynamics models are discussed taking into account random mating of sexes (without formation of permanent male-female couples), possible destruction of the fetus (abortion), and female's pregnancy. One of them deals with both random and directed diffusion in the whole space while in the other the population is assumed to be nondispersing. The population consists of three components: one maleand two female, the latter two being the single (nonfertilized) female and the fertilized one. The case of a separable solution of the limited nondispersing population (in which death moduli can be decomposed into the sum of two terms where one of them depends on time and age and theother is a function of time and the population size) is analyzed. The existence of a unique solution of the Cauchy problem for the nondispersing population model is proved and its longtime behavior is demonstrated. An analogous situation for the dispersing population is analyzed, too.

13

Periodicity of the Population Size of an Age-Dependent Model With a Dominant Age Class by Means of Impulsive Perturbations

Covachev V.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

147-155

EN

For an age-dependent model with a dominant age class an w-periodic regime of the population size is sought by means of impulsive perturbations. For both noncritical and critical cases of first order the problem is reduced to operator systems solvable by a convergent simple iteration method.

14

Optimal Harvesting of the Nonlinear Population Dynamics

Anita S.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

113-122

EN

This paper deals with an optimal harvesting problem for a nonlinear age-dependent population dynamics. The existence and uniqueness of a positive solution for the model considered is demonstrated. The existence of an optimal harvesting effort and the convergence of a certain fractional step scheme are investigated. Necessary optimality conditions for some approximating problems are established.