Research article

Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor

  • In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predator-prey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results.

    Citation: Jing Zhang, Shengmao Fu. Hopf bifurcation and Turing pattern of a diffusive Rosenzweig-MacArthur model with fear factor[J]. AIMS Mathematics, 2024, 9(11): 32514-32551. doi: 10.3934/math.20241558

    Related Papers:

    [1] Li Shen, Jian Liu, Guo-Wei Wei . Evolutionary Khovanov homology. AIMS Mathematics, 2024, 9(9): 26139-26165. doi: 10.3934/math.20241277
    [2] Linlin Tan, Meiying Cui, Bianru Cheng . An approach to the global well-posedness of a coupled 3-dimensional Navier-Stokes-Darcy model with Beavers-Joseph-Saffman-Jones interface boundary condition. AIMS Mathematics, 2024, 9(3): 6993-7016. doi: 10.3934/math.2024341
    [3] Mei Li, Wanqiang Shen . Integral method from even to odd order for trigonometric B-spline basis. AIMS Mathematics, 2024, 9(12): 36470-36492. doi: 10.3934/math.20241729
    [4] Oussama Bouanani, Salim Bouzebda . Limit theorems for local polynomial estimation of regression for functional dependent data. AIMS Mathematics, 2024, 9(9): 23651-23691. doi: 10.3934/math.20241150
    [5] Rolly Czar Joseph Castillo, Renier Mendoza . On smoothing of data using Sobolev polynomials. AIMS Mathematics, 2022, 7(10): 19202-19220. doi: 10.3934/math.20221054
    [6] Yunan He, Jian Liu . Multi-scale Hochschild spectral analysis on graph data. AIMS Mathematics, 2025, 10(1): 1384-1406. doi: 10.3934/math.2025064
    [7] Wan Anis Farhah Wan Amir, Md Yushalify Misro, Mohd Hafiz Mohd . Flexible functional data smoothing and optimization using beta spline. AIMS Mathematics, 2024, 9(9): 23158-23181. doi: 10.3934/math.20241126
    [8] Zhe Su, Yiying Tong, Guo-Wei Wei . Persistent de Rham-Hodge Laplacians in Eulerian representation for manifold topological learning. AIMS Mathematics, 2024, 9(10): 27438-27470. doi: 10.3934/math.20241333
    [9] Raju Doley, Saifur Rahman, Gayatri Das . On knot separability of hypergraphs and its application towards infectious disease management. AIMS Mathematics, 2023, 8(4): 9982-10000. doi: 10.3934/math.2023505
    [10] Chao Wang, Fajie Wang, Yanpeng Gong . Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method. AIMS Mathematics, 2021, 6(11): 12599-12618. doi: 10.3934/math.2021726
  • In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predator-prey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results.



    Abbreviations

    MYA: Million years ago
    MSY: Male specific region of the Y
    XCR: X conserved region
    MSCI: Meiotic sex chromosome inactivation
    XCI: X-chromosome inactivation
    Xp: Paternally inherited X chromosome
    Xi: Inactive X chromosome
    XIC: X Inactivation Center
    RepA: Repeat A
    Rsx: RNA on the silent X

    The class Mammalia (mammals) is divided in to two subclasses: Prototheria (monotremes) and Theria (marsupials and eutherians). All therian mammals have male heterogamety, with an XX female: XY male sex chromosome system (Figure 1), or some simple variant. This sex chromosome system arose before the marsupial/eutherian split, ~ 180 million years ago (MYA) [1], from an ordinary pair of autosomes after a mutation in the Sox3 gene resulted in the birth of the testis-determining gene Sry [2].

    Figure 1. Sex chromosome systems in amniote vertebrates. Eutherian and marsupial mammals both have XY male: XX female sex chromosome systems that share considerable homology (blue). A region that is autosomal in marsupials (red) was added to the X in the eutherian ancestor. Both conserved (blue) and added (red) regions are autosomal in birds and monotremes. In the model monotreme (platypus) males have five X chromosomes and five Y chromosomes, whereas females have five pairs of Xs. Birds have a ZZ male: ZW female sex chromosome system. The bird Z shares homology with platypus X5 (green). The position of Xist and its orthologue (Lnx3) are shown in each lineage. The marsupial specific Rsx is also shown.

    Genes advantageous to males accumulated on the proto Y near Sry either by transposition from autosomal sites or by mutation of existing genes. Recombination with the X was suppressed across this region so that the male advantageous genes were only inherited with the testis-determining gene, giving rise to the male specific region of the Y (MSY). Lack of recombination led to progressive gene loss on, and degradation of, the MSY [3]. Thus, in all therian species the X and Y are morphologically distinct.

    1. Mammal sex chromosomes

    The marsupial X chromosome is ~ 2/3 the size of the eutherian X, and is homologous to the long arm (Xq) and proximal short arm (Xp) of human X chromosome (called the X conserved region; XCR). In contrast, the short arm of the human X chromosome (distal to Xp11.22) is orthologues to marsupial autosomes, so was added to the eutherian X before the radiation of eutherians (~ 105 MYA), but after their divergence from marsupials [4] (Figure 1).

    Monotremes, which diverged from therian mammals ~ 200 MYA, comprise one extant platypus and four extant echidna species, all with a complex of serially translocated sex chromosomes. In the model monotreme, platypus (Ornithorhychus anatinus), males have five X and five Y chromosomes, and females have 5 pairs of X chromosomes [5]. These sex chromosomes do not bear the Sry gene or share homology with the sex chromosome of therian mammals [6] (Figure 1). Instead, they share extensive homology with the independently evolved bird ZW sex chromosome system [6, 7]. Thus, sex chromosomes have evolved multiple times throughout amniote evolution [1] (Figure 1).

    2. Mammal sex chromosome dosage compensation

    In spite of the lethal effect whole chromosome monosomy has for any autosome [8], such grand sex chromosome imbalances are present in many distantly related species. Ohno [9]suggested that copy number imbalance of the X with the autosomes (1X: 2A) in males resulted in the almost twofold upregulation of the X. This led to overexpression from the two Xs in females, which resulted in down-regulation of one X in that sex [10].

    Upregulation of expression from the single X in male is observed in marsupials, where average transcriptional output is near diploid expression levels [11]. However, whether or not the single X is upregulated in male eutherian mammals has remained controversial as a result of inconsistent processing, filtering and analysis methods of transcriptome data [12]. The debate surrounding Ohno’s hypothesis [13, 14, 15, 16, 17, 18] has spawned a novel view that eutherian dosage compensation evolved to restore balance of X genes, which function in protein complexes or protein networks, with their autosomal partners [12]. In some instances expression of X genes were increased to match the original autosomal level (as proposed by Ohno), and in other instances expression of the autosomal genes was decreased to match the new reduced X level This suggests that hyper-expression evolved on a gene-by-gene basis and affected only a subset of X genes.

    In female eutherians and marsupials, down-regulation of X genes to restore parity with the autosomes is achieved by X-chromosome inactivation (XCI); an epigenetic mechanism by which one of the two X chromosomes is silenced in somatic cells. Once silencing has occurred, it is stably maintained throughout all ensuing cell divisions [19]. Although some features that characterize the inactive X (Xi) chromosome are shared between the two lineages, lineage-specific genetic, and epigenetic differences exist [20, 21, 22, 23, 24, 25]. These similarities and differences provide insight into the evolution of mammal XCI.

    3. XCI in therian mammals

    In the male germline of both eutherians and marsupials, sex chromosomes are inactivated during meiosis in a process called meiotic sex chromosome inactivation (MSCI) [26, 27, 28, 29, 30]. All X-borne genes tested in opossum round spermatids were reactivated and expressed [28]. In mouse reactivation is only observed for some genes after spermatogenesis, so the paternally inherited X chromosome (Xp) is delivered to the ovum in a partially pre-inactivated state [31, 32]. After fertilization, transcription of repetitive elements on the Xp is suppressed [33], but biallelic expression is observed for X-borne genes at the two-cell stage [33, 34, 35, 36].

    During eutherian mammal XCI, the choice of which X is to be inactivated can be either random with regard to the parent of origin, or imprinted, where the paternal X is inactivated in all cells. These two forms of XCI are species specific, but can also occur in different cell types within the same species. During mouse pre-implantation development, exclusive silencing of the Xp leads to establishment of imprinted-XCI [37, 38, 39]through to the blastocyst stage. Beyond this, imprinted XCI is maintained only in the trophectodermal extra-embryonic cell lineages that give rise to placental tissue, and the primitive endoderm that gives rise to the visceral endoderm and yolk sac [40]. In contrast, in the developing inner cell mass, which gives rise to the embryo proper, the inactive Xp is reactivated, which is then followed by random XCI [40, 41, 42]. Imprinted XCI was also observed in extra-embryonic cell lineages of rat [43] and cow [44, 45]. However, in human, monkey, horse, pig, mule and rabbit random XCI was observed in both embryonic and extra-embryonic cells [46, 47].

    One of the marked differences between marsupial and eutherian XCI is the choice of X to be inactivated in the embryo proper. XCI in marsupial extra-embryonic, fetal, and adult tissues is imprinted, with the paternally derived X always silenced [24, 25, 48]. The reason and cause for this difference in choice during X inactivation is not understood.

    Although the marsupial inactive X shares some similarities with the eutherian Xi at the cytogenetic level, such as late replication at S phase and heterochromatinization [49, 50, 51, 52, 53, 54], it differs at the molecular level [22, 25]. There are considerable differences in the histone profile of the inactive X between eutherians and marsupials [21] (Table 1), but in general at the onset of XCI, the inactive X loses epigenetic modifications associated with active transcription (e.g. H3K9ac, H4Kac and H3K4me2) and sequentially acquires repressive marks characteristics of silenced chromatin (e.g. H4K20me1 and H3K27me3). In addition, the eutherian Xi exhibits enrichment of histone variants such as macro-H2A, and hypermethylation of promoter sequences stabilizes inactivation once repression has occurred [55] (Table 1). The Xi in marsupial female possum and potoroo metaphase appears hypomethylated [23], in addition promoter DNA methylation appears absent on the Xi for loci tested in opossum [25, 56].

    Table 1. Comparison of repressive and active epigenetic marks on the inactive X chromosome in eutherian (mouse, human) and marsupials (Opossum).
     | Show Table
    DownLoad: CSV

    4. Genes that escape inactivation

    The final outcome of these modifications is silencing of transcription from most genes on the Xi. However, some genes escape inactivation, and as a result are expressed from both active and inactive X chromosomes [57]. The chromatin state of these genes more closely resembles that of expressed genes on the active X and autosomes, than that of silent Xi loci. The number and identity of genes that escape inactivation is different between species. In human somatic cells 15% of genes on the X escape inactivation [58, 59, 60], with a higher frequency on the short arm (orthologues to marsupial autosomes) than on the long arm of the human X (homologous to the marsupial X). Furthermore, about 10% of human X-borne genes have variable inactivation status between tissues and/or individuals [60, 61]. In mouse somatic cells, almost all X-borne genes are inactivated; only 3% escape [59, 62].

    In female mouse trophoblast stem cells [63] and extra-embryonic endoderm [64], both of which are subject to imprinted XCI, a larger number of X-borne genes (13% and 15%) are expressed from both X chromosomes. However, different subsets of genes in these extra-embryonic cell lineages are subject to XCI. The inactive Xp in mouse extra-embryonic tissues globally accumulates the same repressive histone marks as the Xi in other somatic cell types [36, 64, 65]. However, the order in which these modifications appear on the Xi is different. In random XCI enrichment of macro-H2A is a late stage event. In contrast, during imprinted XCI enrichment of H3K27me3 and macro-H2A appear early on the Xi, whereas H3K9me2 accumulation is detected later [42, 66]. Similar to Xi in somatic cells, X-borne promoters are hypermethylated on the inactive X chromosome [67].

    In marsupials, a large proportion of X-borne genes escape XCI [20, 25], as in the mouse extra-embryonic membranes. Genes on the marsupial Xi exhibit variable levels of incomplete silencing across species, tissue and developmental stage [68, 69]. Approximately 15% of genes on the American grey short-tailed opossum X escape inactivation [25]. As such imprinted XCI in marsupials and mouse extra-embryonic tissues is not as complete as random XCI in eutherian somatic tissue. Interestingly, the marsupial Xi lacks the repressive H4K20me1 mark [21], which accumulates on Xi during both imprinted and random XCI in mouse [70] (Table 1). Localization of macro-H2A to the marsupial Xi is unknown.

    5. X Inactivation Center (XIC)

    The silencing achieved during XCI is triggered by long-noncoding (lnc) RNAs that interact with chromatin regulatory complexes to alter chromosome conformation. Yet despite the central role of RNA-chromatin interactions during XCI, they are not fully understood.

    A region on the eutherian X chromosome called the X Inactivation Centre (XIC) is of key importance in coordinating XCI. The XIC contains several pseudogenes (e.g. Fxyd6) and protein-coding genes (e.g. CDX4, CHIC1, SLC16A2) [71], along with the key non-coding RNA genes (e.g. XIST, TSIX, FTX, JPX and others) (Figure 2). The XIC lncRNAs are poorly conserved between eutherian species, with the master regulator (XIST) the most conserved element between sequenced eutherian genomes [72, 73, 74]. However, there is no ortholog of XIST in the marsupial or monotreme genomes, in which the XIC locus has been disrupted [72, 75]. Interestingly, in chicken the locus remains intact with protein coding genes that share homology to XIST and the mouse Tsx gene [73].

    Figure 2. Comparative maps of the X inactivation center (XIC) in mouse and orthologous region in chicken. The mouse XIC spans 8cM (10‒20 Mb) and bears several non-coding RNAs as well as protein-coding genes. Only elements around Xist are shown. Arrows identify direction of each transcription unit. Protein coding genes are indicated in white and genes producing lncRNAs are blue. Genes with imprinted expression in mouse extra-embryonic tissues are marked by an asterisk, and the expressed allele is indicated (i.e. p = paternally expressed, m = maternally expressed). Putative positive regulators of Xist (Jpx, Ftx, Rnf12) are labeled in green, and putative negative regulators (Linx, Xite, Tsix) are labeled in red. Lines identify homologous genes in chicken.

    6. Evolution of the master regulator—XIST

    Several exons of the chicken protein-coding gene Lnx3 share homology with the XIST gene [73, 76, 77] (Figure 3). These homologies reveal that the XIST promoter evolved from exons 1 and 2 of the Lnx3 gene, which is among the most conserved regions of the XIST gene between different eutherian species [73, 76, 77]. The remaining XIST exons (that share no homology with Lnx3) are likely to have originated via transposition of various mobile elements, presumably endogenous retroviruses, fragments of which were amplified to generate several simple tandem repeats [76, 78]. The lack of XIST in marsupials, along with it being in all eutherian genomes, means that XIST evolved as a key player in XCI in the eutherian ancestor.

    Figure 3. Homology between the eutherian Xist, and the protein coding Lnx3 gene in chicken. Human and mouse Xist both have eight exons. Functional domains of Xist include the tandem repeats (labeled A to F). The Xist promoter (P) originates from exons 1 and 2 of Lnx3. Colors shared between chicken Lnx3 and human/mouse Xist, identify homologies and, therefore, origin of Xist exons. Yellow Xist exons originated from mobile elements. Human exons are numbed h1‒h8, and mouse exons m1‒m8. Homology with exon 3 of Lnx3 is detectable in the human and mouse genome, but gives rise to pseudo-exons (not shown).

    The ancestral XIST gene presumably consisted of ten exons [74, 76]. Two large exons (1 and 8) together constitute about 90% of XIST and contain tandem repeats (denoted A to H) [74] (Figure 3). Tandem repeat A is conserved in all eutherian species [75, 76, 78, 79, 80, 81, 82, 83, 84], whereas presence, absence, or amplification of the other repeats is species specific [76, 83]. Six of the ancestral exons are conserved across Eutheria [73, 76, 77, 78], with the remaining four (2, 6, 9, 10) either functional- or pseudo-exons depending on species [74, 85]. Thus, human XIST encodes a 19kb transcript [86], whereas the mouse Xist transcript is 17 kb [87]. Despite variable intron-exon structures between species, XIST exons are GC dinucleotide rich compared to the introns. The proportion of GC richness is constant between species (39.8-42.2%) and similar to the whole XIC locus.

    7. The role of Xist

    In mouse, Xist is transcribed only in females by RNA polymerase II, solely from the Xi. Analysis of the CpG dinucleotide methylation patterns in the promoter region has shown that the active Xist allele (on the inactive X) is completely unmethylated [88]. In contrast, the silent maternal Xist allele is fully methylated [88]. Although Xist RNA is spliced and polyadenylated, it is absent from polysomes [80, 89] and remains in the nucleus where it coats and forms a “Xist cloud” on the X to be inactivated [80, 90], the spreading of Xist RNA along one X chromosome in cis initiates the chromosome-wide silencing.

    LINE1 retrotransposons are enriched on the X (mouse X ~ 28.5%, autosomes ~ 14.6%) [91], so were proposed to be anchor points for Xist to ensure efficient spreading of the machinery responsible for silencing [92]. A significant decrease in LINE1 density at regions containing genes that escape inactivation [93]supports this hypothesis, although a less direct role for LINEs in the spreading process seems more probable [94]. Accordingly, LINEs were proposed to moderate spatial organization of the transcriptionally silent nuclear territory of the inactive X chromosome, into which X-borne genes are recruited as they are silenced [95, 96].

    Xistexpression is followed by the formation of a repressive chromatin state that excludes transcriptional machinery from the inactive X [95]. Repeat A (RepA), at the 5' end of Xist, recruits the polycomb repressive complexes PRC1 and PRC2 to the Xi. Polycomb repressive complexes decorate the Xi and catalyze the characteristic repressive histone modifications of Xi. A number of other proteins are also localized to the Xi, potentially trafficked via Xist RNA, including nuclear scaffolding factors such as SAF-A [97] and the histone variant macro-H2A [98, 99, 100, 101].

    8. Activators and repressors of XIST

    During the morula/blastocyst stage in mice, a few days after initiation of imprinted XCI, Tsix is expressed exclusively from the maternally derived X chromosome to inhibit expression from the maternally derived Xist [102, 103, 104]. However, during random XCI Tsix demarcates whichever X remains active (Xa) [105, 106, 107] and its expression prevents in cis transcription of Xist and, ultimately, inactivation of that X [104]. Tsix in rodents spans more than a 40 kb region that encompasses the entire transcription unit of Xist [108]. In primates, cow and dog there are many species-specific repeat-element insertions, and large deletions, that disrupt the overall structure of TSIX [109]. In human, TSIX appear to be an expressed pseudogene unable to repress XIST, and overlaps only with the last two exons of XIST [109, 110]. In contrast to mouse Tsix, the human TSIX is expressed with XIST from the Xi in the fetal cells, throughout gestation, but cease transcription shortly after birth [110]. Thus, Tsix function seems limited to rodents.

    Ftx and Jpx potentially upregulate Xist, and are conserved in mouse, human, and cow [71, 111], evolving from the protein coding genes Wave4 and Uspl, respectively [76]. Both genes escape imprinted XCI and are expressed predominantly from the paternal allele at the pre-implantation stage [112]. Deletion of the Ftx promoter leads to decreased Xist expression in male embryonic stem (ES) cells [111],indicating that it is a positive regulator of Xist. However a recent study shows that Ftx disruption did not affect embryo survival, or expression of Xist and other X-borne genes during pre-implantation, thus is dispensable for imprinted inactivation [113]. Whether Ftx is involved in random XCI in post-implantation embryos is yet to be determined.

    Jpx is located just downstream of the Ftx locus, and approximately 10 kb upstream of Xist [71, 114, 115]. Jpx escapes XCI and can upregulate Xist expression on the Xi [71, 115, 116] by evicting CTCF from the Xist promoter [117]. Deletion of a single Jpx allele in XX female ES cells results in failed accumulation of Xist on either X, and inactivation is prevented [116].

    Although the function of XIST may be well conserved in eutherians, other elements of the XIC (even those with sequence conservation) may not have conserved function (Table 2). However, poor sequence conservation of noncoding RNAs in the XIC does not necessarily indicate a lack of function [118, 119], as maintaining secondary structure (and therefore function) of lncRNA molecules may only require short stretches of sequence preservation. This poor conservation might indicate their adaptation to function in specific genomic environments, suggesting that regulation of XCI is at least partially species-specific. For instance, the recently evolved lncRNA gene XACT is only present in human and chimpanzee, but not in macaques or more distantly related species [120]. XACT is expressed from and coats the active X in female human embryonic stem cells and early differentiating cells, and may contribute to protecting the Xa from inactivation [120].

    Table 2. Sequence conservation of different genes involved in XCI between eutherian (mouse and human) and marsupials (Opossum).
    Eutherian Marsupial
    Mouse Human Opossum
    lncRNAs Xist ×
    Ftx ×
    Jpx ×
    Tsix * ×
    XACT × ×
    Rsx × ×
    √ = presence, × = absence, * = pseudogene.
     | Show Table
    DownLoad: CSV

    9. A marsupial XIST like gene: Rsx

    The marsupial Lnx3 (the precursor of XIST) gene has a native open reading frame that is expressed in both males and females. The eutherian XIC locus is disrupted in marsupials, and Lnx3 presumably functions as a PDZ domain containing ring finger protein rather than an untranslated nuclear RNA similar to XIST. Consequently, the X inactivation process in marsupials involves neither XIST nor the XIC.

    The marsupial X chromosome has multiple large-scale internal rearrangements with respect to both the human X, and between Australian and American representative species [121]. This is contrary to the generally conserved gene order on the eutherian X [122, 123], presumably due to purifying selection against rearrangements that perturb interactions between XIST and regions of the X intended for inactivation [93]. Extensive rearrangement of the marsupial X chromosome was taken as support for the lack of a XIST equivalent [121]. However, Rsx (RNA on the silent X) was identified in opossum (and two Australian marsupials), and appears to fulfill some of the functions of XIST [22]. As such, the epigenetic mechanisms that silence the inactive X in the somatic cells of marsupials and eutherians share a rema rkable degree of convergence.

    The mature Rsx in opossum is a 27-kb non-coding RNA with several XIST-like characteristics, such as a high GC content and enrichment of conserved 5’ tandem repeats that may be involved in the formation of stem-loop structures [22]. These are potentially important functional domains necessary for directing protein complexes responsible for chromatin modification that repress transcription. However, further studies are needed to determine the candidate functional domains of Rsx.

    Rsx is located adjacent to Hprt on the marsupial X in a different genomic context to—and shares no sequence homology with—XIST. Yet, like XIST, Rsx is expressed exclusively in female somatic cells [22] and extra embryonic membranes [25], but not in germ cells where both X chromosomes are active [22]. Rsx is expressed in cis from Xi, around which it forms a “Rsx-cloud” that results in repressed gene activity. Moreover, after introduction of Rsx into mouse ES cells, Rsx RNA coated the transgenic chromosome and resulted in its inactivation in more than half the cells examined [22].

    Monoallelic expression from the paternally derived allele of Rsx, in both fetal brain and extra embryonic membranes, was shown to be due to different epigenetic characteristics of the active and inactive alleles. Rsx, similar to Xist, is differentially methylated at its promoter. There is ~ 100% methylation of the maternally derived allele, and virtually no methylation of the paternally derived allele [25]. Furthermore, H3K27me3 repressive mark was absent from the Rsx gene body, demonstrating that similar epigenetic mechanisms regulate the independently evolved Rsx and XIST genes [25].

    10. Conclusion

    Although independently evolved, there appear to be remarkable functional similarities shared by Xist and Rsx. However, it remains unknown if Rsx can perform all functions attributed to Xist, or if it traffics the epigenetic machinery as Xist is proposed to do. Since overlapping, but different, suites of repressive chromatin modifications are used to silence the X in eutherians and marsupials, many of these epigenetic tools were likely utilized in the therian ancestor to achieve X chromosome inactivation. However, it is yet to be determined if Rsx was an XCI switch in the therian mammal ancestor that was retained in marsupials, and then replaced by Xist in the eutherian ancestor; or if they evolved simultaneously in the two lineages. Perhaps the epigenetic differences observed between eutherian and marsupial XCI merely reflect that these two lncRNAs direct protein complexes that are responsible for different chromatin modification. Finally, the potential existence of a marsupial X inactivation center close to Rsx, which bears lncRNAs that may regulate Rsx expression, remains a fascinating possibility.

    Acknowledgments

    P.D.W. was supported by an ARC fellowship.

    Conflict of Interest

    The authors declare no conflicts of interest in this paper.



    [1] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385–387. https://doi.org/10.1126/science.171.3969.385 doi: 10.1126/science.171.3969.385
    [2] M. E. Gilpin, M. L. Rosenzweig, Enriched predator-prey systems: Theoretical stability, Science, 177 (1972), 902–904. https://doi.org/10.1126/science.177.4052.902 doi: 10.1126/science.177.4052.902
    [3] R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900–902. https://doi.org/10.1126/science.177.4052.900 doi: 10.1126/science.177.4052.900
    [4] M. R. Myerscough, M. J. Darwen, W. L. Hogarth, Stability, persistence and structural stability in a classical predator-prey model, Ecol. Model., 89 (1996), 31–42. https://doi.org/10.1016/0304-3800(95)00117-4 doi: 10.1016/0304-3800(95)00117-4
    [5] A. Persson, L. A. Hansson, C. Brönmark, P. Lundberg, L. B. Pettersson, L. Greenberg, et al., Effects of enrichment on simple aquatic food webs, Am. Nat., 157 (2001), 654–669. https://doi.org/10.1086/320620 doi: 10.1086/320620
    [6] M. Vos, B. W. Kooi, D. L. DeAngelis, W. M. Mooij, Inducible defences and the paradox of enrichment, Oikos, 105 (2004), 471–480. https://doi.org/10.1111/j.0030-1299.2004.12930.x doi: 10.1111/j.0030-1299.2004.12930.x
    [7] M. L. Rosenzweig, R. H. MacAhthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209–223. https://doi.org/10.1086/282272 doi: 10.1086/282272
    [8] P. Turchin, Complex population dynamics, Princeton: Princeton University Press, 2003.
    [9] G. F. Fussmann, B. Blasius, Community response to enrichment is highly sensitive to model structure, Biol. Lett., 1 (2005), 9–12. https://doi.org/10.1098/rsbl.2004.0246 doi: 10.1098/rsbl.2004.0246
    [10] G. Seo, G. S. K. Wolkowicz, Sensitivity of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: A bifurcation theory approach, J. Math. Biol., 76 (2018), 1873–1906. https://doi.org/10.1007/s00285-017-1201-y doi: 10.1007/s00285-017-1201-y
    [11] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/ent91293-5 doi: 10.4039/ent91293-5
    [12] V. S. Ivlev, Experimental ecology of the feeding of fishes, New Haven: Yale University Press, 1961.
    [13] A. D. Jassby, T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnol. Oceanogr., 21 (1976), 540–547. https://doi.org/10.4319/lo.1976.21.4.0540 doi: 10.4319/lo.1976.21.4.0540
    [14] L. Y. Zanette, A. F. White, M. C. Allen, C. Michael, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398–1401. https://doi.org/10.1126/science.1210908 doi: 10.1126/science.1210908
    [15] X. Y. Wang, L. Zanette, X. F. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. https://doi.org/10.1007/s00285-016-0989-1 doi: 10.1007/s00285-016-0989-1
    [16] Y. Wang, X. F. Zou, On a predator-prey system with digestion delay and anti-predation strategy, J. Nonlinear Sci., 30 (2020), 1579–1605. https://doi.org/10.1007/s00332-020-09618-9 doi: 10.1007/s00332-020-09618-9
    [17] H. S. Zhang, Y. L. Cai, S. M. Fu, W. M. Wang, Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comput., 356 (2019), 328–337. https://doi.org/10.1016/j.amc.2019.03.034 doi: 10.1016/j.amc.2019.03.034
    [18] J. Wang, Y. L. Cai, S. M. Fu, W. M. Wang, The effect of the fear factor on the dynamics of a predator-prey model incorporating the prey refuge, Chaos, 29 (2019), 083109. https://doi.org/10.1063/1.5111121 doi: 10.1063/1.5111121
    [19] T. Qiao, Y. L. Cai, S. M. Fu, W. M. Wang, Stability and Hopf bifurcation in a predator-prey model with the cost of anti-predator behaviors, Int. J. Bifurcat. Chaos, 29 (2019), 1950185. https://doi.org/10.1142/s0218127419501852 doi: 10.1142/s0218127419501852
    [20] S. Mondal, G. P. Sudeshna, Dynamics of a delayed predator-prey interaction incorporating nonlinear prey refuge under the influence of fear effect and additional food, J. Phys. A, 53 (2020), 295601. https://doi.org/10.1088/1751-8121/ab81d8 doi: 10.1088/1751-8121/ab81d8
    [21] Y. Shi, J. H. Wu, Q. Cao, Analysis on a diffusive multiple Allee effects predator-prey model induced by fear factors, Nonlinear Anal. Real World Appl., 59 (2021), 103249. https://doi.org/10.1016/j.nonrwa.2020.103249 doi: 10.1016/j.nonrwa.2020.103249
    [22] P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, Stability and bifurcation analysis of a three-species food chain model with fear, Int. J. Bifurcat. Chaos, 28 (2018), 1850009. https://doi.org/10.1142/s0218127418500098 doi: 10.1142/s0218127418500098
    [23] P. Panday, N. Pal, S. Samanta, J. Chattopadhyay, A three species food chain model with fear induced trophic cascade, Int. J. Appl. Comput. Math., 5 (2019), 100. https://doi.org/10.1007/s40819-019-0688-x doi: 10.1007/s40819-019-0688-x
    [24] W. J. Ni, J. P. Shi, M. X. Wang, Global stability of spatially nonhomogeneous steady state solution in a diffusive Holling-Tanner predator-prey model, Proc. Amer. Math. Soc., 149 (2021), 3781–3794. https://doi.org/10.1090/proc/15370 doi: 10.1090/proc/15370
    [25] R. Yafia, M. A. Aziz-Alaoui, Existence of periodic travelling waves solutions in predator prey model with diffusion, Appl. Math. Model., 37 (2013), 3635–3644. https://doi.org/10.1016/j.apm.2012.08.003 doi: 10.1016/j.apm.2012.08.003
    [26] M. Banerjeea, V. Volpert, Spatio-temporal pattern formation in Rosenzweig-MacArthur model: Effect of nonlocal interactions, Ecol. Complex., 30 (2017), 2–10. https://doi.org/10.1016/j.ecocom.2016.12.002 doi: 10.1016/j.ecocom.2016.12.002
    [27] S. M. Fu, H. S. Zhang, Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105807. https://doi.org/10.1016/j.cnsns.2021.105807 doi: 10.1016/j.cnsns.2021.105807
    [28] X. Li, W. H. Jing, J. P. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA J. Appl. Math., 78 (2013), 287–306. https://doi.org/10.1093/imamat/hxr050 doi: 10.1093/imamat/hxr050
    [29] J. D. Meiss, Differential dynamical systems, Philadelphia: Society for industrial and applied mathematics, 2007. https://doi.org/10.1137/1.9780898718232
    [30] L. Perko, Differential equations and dynamical systems, New York: Springer, 1996.
    [31] B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and applications of Hopf bifurcation, Cambridge-New York: Cambridge University Press, 1981.
    [32] H. B. Shi, S. G. Ruan, Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534–1568. https://doi.org/10.1093/imamat/hxv006 doi: 10.1093/imamat/hxv006
    [33] X. Y. Gao, S. Ishag, S. M. Fu, W. J. Li, W. M. Wang, Bifurcation and Turing pattern formation in a diffusive ratio-dependent predator-prey model with predator harvesting, Nonlinear Anal. Real World Appl., 51 (2020), 102962. https://doi.org/10.1016/j.nonrwa.2019.102962 doi: 10.1016/j.nonrwa.2019.102962
    [34] Y. L. Song, T. H. Zhang, Y. H. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci. Numer. Simul., 33 (2016), 229–258. https://doi.org/10.1016/j.cnsns.2015.10.002 doi: 10.1016/j.cnsns.2015.10.002
    [35] Y. Lou, W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equ., 131 (1996), 79–131. https://doi.org/10.1006/jdeq.1996.0157 doi: 10.1006/jdeq.1996.0157
    [36] Y. F. Jia, Y. Li, J. H. Wu, Effect of predator cannibalism and prey growth on the dynamic behavior for a predator-stage structured population model with diffusion, J. Math. Anal. Appl., 449 (2017), 1479–1501. https://doi.org/10.1016/j.jmaa.2016.12.036 doi: 10.1016/j.jmaa.2016.12.036
    [37] K. Ryu, I. Ahn, Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics, J. Math. Anal. Appl., 283 (2003), 46–65. https://doi.org/10.1016/s0022-247x(03)00162-8 doi: 10.1016/s0022-247x(03)00162-8
    [38] E. N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131–151. https://doi.org/10.1016/0022-247x(83)90098-7 doi: 10.1016/0022-247x(83)90098-7
    [39] L. Li, On positive solutions of a nonlinear equilibrium boundary value problem, J. Math. Anal. Appl., 138 (1989), 537–549. https://doi.org/10.1016/0022-247x(89)90308-9 doi: 10.1016/0022-247x(89)90308-9
    [40] J. K. Hale, H. Kocak, Dynamics and bifurcations, New York: Springer, 1991. https://doi.org/10.1007/978-1-4612-4426-4
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(793) PDF downloads(54) Cited by(0)

Article outline

Figures and Tables

Figures(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog