Introduction

Various aquatic ectotherms species are exploited in artisanal and industrial fisheries (Arreguín-Sánchez and Arcos-Huitrón, 2011; Malcolm et al. 2021). Given their sensitivity to temperature changes, many of these species and consequently their fisheries may be seriously affected by the effects of climate change (Audzijonyte et al. 2016; Pinsky et al. 2019; Jorgensen et al. 2022). Quantifying and forecasting these impacts are necessary tasks for the protection and future management of these fisheries.

At the regional level, knowing the thermal preferences (temperature range for proper ectotherm development) of a capture species, as Octopus maya, allows analyzing important aspects of its ecophysiology and geographic distribution. Consequently, estimating the impacts on artisanal fishing communities enables food security that depends on their fishing activity to a large extent (Salas et al. 2022).

Food security is the state of well-being in which people have safe food available at all times to meet their nutritional needs and can access it by physical and economic means (FAO 2008). Since many artisanal fisheries are subsistence fisheries, they are directly related to the food security of the fishermen who practice them (Salas et al. 2022). The capture of the endemic benthic O. maya in the Yucatan coasts sustains the octopus fishery in Yucatan, particularly important for the families of artisanal fishermen, since a good part of their annual income depends on a good fishing season of this species (Salas et al. 2022).

Given the great sociocultural importance of the O. maya fishery, numerous studies have been conducted on this species in terms of its physiology (Noyola et al. 2013a; Juárez et al. 2015; López-Galindo et al. 2019), distribution (Gamboa-Álvarez et al. 2015; Ángeles-González et al. 2021a; Ángeles-González et al. 2021b), and management (Duarte et al. 2018; Rosales Raya and Fraga Berdugo 2019). It is important to point out that since O. maya is an ectotherm with benthic habits and high endemism, its natural population could be dangerously threatened by the increase in sea temperature due to climate change (Ángeles-González et al. 2021a).

In particular, several studies have suggested that increased temperature may inhibit egg laying (Juárez et al. 2015) and reduce the number of eggs laid (Juárez et al. 2015), decrease the size at hatching (Caamal-Monsreal et al. 2016), produce physiologically damaged embryos (Caamal-Monsreal et al. 2016), reduce biomass accumulation of juveniles (Noyola et al. 2013b; Escamilla-Aké et al. 2023) and sexual fitness (López-Galindo et al. 2019), increase mortality at all life stages (Juárez et al. 2015) with damage transmission to progeny (Vargas-Abúndez et al. 2023), and modify their geographic distribution (Ángeles-González et al. 2021a). However, incorporating physiological information has been a challenge for this species into descriptive and predictive models of the spatial and temporal dynamics of individuals and populations for conservation and management purposes.

Despite the efforts to describe abundance patterns with the season and temperature (Gamboa-Álvarez et al. 2015; Avendaño et al. 2019; Ángeles-González et al. 2021a; Avendaño et al. 2022), currently information on O. maya populations dispersal and propagation patterns still lacking. Consequently, the ability to prescribe the management of this fishery is limited. This situation acquires greater relevance in shallow waters (depth < 30 m) because these are the areas exploited by artisanal fishermen, between 10 and 30 m depth (Salas et al. 2019a) with an average of 12 m (Salas et al. 2009). In this respect, the integrodifference equation (IDE) seems an alternative to fill this need.

The IDE is a tool that allows incorporating growth and population dispersion in a continuous space but in discrete time (for example time steps of years or generations), as well as estimating the speed at which a population spreads in space (Lutscher 2019). This equation is widely used in mathematical biology and especially in theoretical ecology (Zhou and Kot 2011; Otto et al. 2022), showing versatility and a great variety of applications from the description of invasive species (Deeley and Petrovskaya 2022) to the species interaction in dynamics and heterogeneous environments (Latore et al. 1999; Pearce et al. 2006). Thus, the IDE could be a powerful tool for modeling the relative persistency, dispersal, and spread of O. maya natural population.

At the current temperatures, O. maya is relatively abundant in shallow waters (Gamboa-Álvarez et al. 2015), although sizes tend to be smaller than in deeper waters (depth > 30 m) (Avendaño et al. 2019). However, as temperature increases, the species is expected to show various adaptive behaviors, ranging from acclimatization (Vargas-Abúndez et al. 2023) to migration (Angeles-Gonzalez et al. 2017). Such behaviors influence reproductive patterns (Angeles-Gonzalez et al. 2017) and biomass accumulation on a spatial and temporal scale (Avendaño et al. 2019; Gamboa-Álvarez et al. 2015). Changes in time and space in O. maya populations (Avendaño et al. 2022) may modify the fishing dynamics of this species and, consequently, trigger mitigation and adaptation mechanisms at the regional level, especially in communities where artisanal fishing predominates (Salas et al. 2019b).

Arreguín-Sánchez (2019) suggests that warming has been and will keep increasing O. maya catches in the Campeche Bank. However, this suggestion contrasts with several recent experimental results that have detected negative temperature effects on both juveniles and adults above 27 °C (Juárez et al. 2015; López-Galindo et al. 2019; Meza-Buendía et al. 2021). Although temperature tends to accelerate some vital rates, such as growth and maturation, it does not do so indefinitely, since it is limited to temperature intervals and exposure times that are tolerable for the species (Pascual et al. 2019; Escamilla-Aké et al. 2023).

Over the next few years, laboratory results and conjectures about increasing O. maya catches with increasing temperature are compatible below a specific temperature and will be largely due to an increase in the proportion of adults rather than an assumed benefit of elevated temperatures for population growth of this species. However, these two opposing positions may be compatible below a given temperature if we take into account the change in population configuration.

Finally, it remains to be clarified how temperature might evolve until the end of the century. In that sense, shared socioeconomic pathways (SSPs) can be useful frameworks for understanding how temperature will increase and its impacts on shallow-water O. maya fisheries.

With this background, this paper formulates mathematically the vital rates of survival, maturation, and reproduction of O. maya according to its thermal preferences in its immature and mature stages in order to estimate the impact of the increase in ocean temperature due to climate change on the population of O. maya and the food security of artisanal fishermen from the Yucatan coast who catch this species in shallow waters of the Yucatan shelf.

Material and methods

Shared socioeconomical pathways

The SSPs describe plausible alternative changes in aspects of society, such as demographic, economic, technological, social, governmental, and environmental factors that could hinder or facilitate climate change mitigation or adaptation in this century without considering climate change per se or assuming climate policies per se (O’Neill et al. 2014). Considering the different challenges for climate change mitigation or adaptation, the SSPs: SSP1, SSP2, SSP3, SSP4, and SSP5 are used as a frame of reference through their projected global temperature increases for the year 2100 (Riahi et al. 2017); thus, plausible contexts are described or the O. maya fishery and artisanal fishermen who catch this species in shallow waters.

The SSP1 is the “sustainability” narrative scenario, with few challenges for mitigation and adaptation; the SSP2 is the “middle of the road” scenario, with moderate challenges for mitigation and adaptation; the SSP3 is the “regional rivalry” scenario, with large challenges for mitigation and adaptation; the SSP4 is the “inequality” scenario, with few challenges for mitigation but large challenges for adaptation; and the SSP5 is the “fossil-fueled development” scenario, with large challenges for mitigation and adaptation (O’Neil et al. 2017). By 2100, global temperature increases of approximately 3 °C are projected for SSP1; 4 °C for the SSP2, SSP3, and SSP4; and 5 °C for SSP5 (Riahi et al. 2017).

Temperature and abundance of Octopus maya in the study area

The northern Yucatan shelf up to 250 km wide has a slight slope that adds approximately 1 m in depth for every 2 km offshore until just before 100 km; a relatively flat zone of about 40 km follows, ending with a deeper zone that adds approximately 1.5 m for every 2 km offshore (Merino 1991).

The water near the coast is warmer, decreasing in temperature as it moves away from the coast and increasing in depth and less warm to the east due to upwelling (Merino 1991; Enriquez et al. 2013). This reduction in temperature northeastward is more pronounced at the beginning of the upwelling phenomenon that occurs in spring, decreasing gradually until the end of summer (Merino 1991). In autumn, the upwelling effects are reduced, so temperature decreases from the coast toward the north, mainly due to depth (Merino 1991). In winter, winds play an important role in homogenizing the shelf temperature, reducing the differences in the temperature gradient due to the depth (Merino 1991).

So far, O. maya is abundant in shallow waters off the Yucatan Shelf up to a maximum depth of 60 m (Avendaño et al. 2019). Off the coast of Yucatan, up to 10 m in depth, the bottom temperature throughout the year varies from 22 to 30 °C with an average of 26 °C (Enriquez et al. 2013). Between 10 and 30 m in depth, the average bottom temperature is 23 °C in winter, 26 °C in autumn, 25 °C in summer, and 23 °C in spring (Merino 1991), which approximately give an annual average temperature of 24 °C. Up to 30 m, shallow water areas are exploited by artisanal fishermen (Salas et al. 2009), while depths greater than 30 m are exploited by semi-industrial fleets (Avendaño et al. 2019; Salas et al. 2019a).

Population growth in shallow waters

The presence \(f(T;{T}_{i},{\sigma }_{i})\) of O. maya at a given temperature \(T\) was assumed to have a normal distribution, with mean \({T}_{i}\) and standard deviation \({\sigma }_{i}\) defined by the preferred temperature and thermal preference, respectively, which this species has shown experimentally at each substage \(i\) (juvenile and adult) of its life.

$$f\left(T;{T}_{i},{\sigma }_{i}^{2}\right)=\frac{1}{{\sigma }_{i}\sqrt{2\pi }}{\text{exp}}\left(-\frac{{\left[T-{T}_{i}\right]}^{2}}{2{\sigma }_{i}^{2}}\right)$$
(1)

where O. maya thermal preferences are estimated as \({T}_{j}=23.4^\circ C\) (Noyola et al. 2013a) for juveniles (immature \({j}_{1}\) and mature \({j}_{2}\)) and \({T}_{a}=24^\circ C\) (Rosas et al. 2014) for male (\({a}_{1}\)) and female \(\left({a}_{2}\right)\) adults. According to O. maya thermal preferences of in each stage, standard deviations are empirically estimated as \({\sigma }_{{j}_{1}}=6^\circ C\) for immature juveniles \({j}_{1}\), and \({\sigma }_{{j}_{2}}={\sigma }_{{a}_{1}}=3^\circ C\) for mature juveniles \({j}_{2}\) and for breeding males \({a}_{1}\), and \({\sigma }_{{a}_{2}}=1^\circ C\) for breeding females \({a}_{2}\) (Rosas et al. 2014).

The presence of O. maya in each stage at temperature \(T\) with respect to the maximum presence at a preferred temperature \({T}_{i}\) was used to estimate temperature-dependent vital rates: adult survival fraction \({s}_{a}(T)\), juvenile survival fraction \({s}_{j}(T)\), fraction of mature juveniles \(g(T)\), and offspring number per breeding adult \({r}_{a}(T)\). Vital rates formed the primitive matrix \(B\left(T\right)\) that modulates the growth of a population stratified into juveniles and adults without considering dispersion.

$$\left[\begin{array}{c}{J}_{t+1}\\ {A}_{t+1}\end{array}\right]=\left[\begin{array}{cc}{s}_{j}\left(T\right)\left[1-g\left(T\right)\right] & {r}_{a}\left(T\right)\\ {s}_{j}\left(T\right)\cdot g\left(T\right)& \frac{1}{2}{\cdot s}_{a}\left(T\right)\end{array}\right]\left[\begin{array}{c}{J}_{t}\\ {A}_{t}\end{array}\right]$$
(2)

where explicit vital rates are given by:

$${s}_{a}\left(T\right)={\text{exp}}\left(-\frac{{\left[T-{T}_{{a}_{1}}\right]}^{2}}{2{\sigma }_{{a}_{1}}^{2}}\right)$$
(3.a)
$${s}_{j}\left(T\right)={\text{exp}}\left(-\frac{{\left[T-{T}_{{j}_{2}}\right]}^{2}}{2{\sigma }_{{j}_{2}}^{2}}\right)$$
(3.b)
$$g\left(T\right)={\text{exp}}\left(-\frac{{\left[T-{T}_{{j}_{1}}\right]}^{2}}{2{\sigma }_{{j}_{1}}^{2}}\right)$$
(3.c)
$${r}_{a}\left(T\right)=\frac{1}{2}\cdot {\rho }_{\text{max}}\cdot {\text{exp}}\left(-\frac{{\left[T-{T}_{{a}_{2}}\right]}^{2}}{2{\sigma }_{{a}_{2}}^{2}}\right)$$
(3.d)

where \({\rho }_{\text{max}}\) is the maximum number of offspring for female. In this study, \({\rho }_{\text{max}}=2500\).

Dispersion in shallow waters

Dispersion was incorporated with an IDE:

$${{\varvec{N}}}_{t+1}\left(x\right)=\underset{\Omega }{\overset{\hspace{0.17em}}{\int }}K\left(x-y,T\right)\circ B\left(T\right){{\varvec{N}}}_{t}\left(y\right)dy$$
(4)

where \(\circ\) denotes the Hadamard product of matrices, with a symmetric Gaussian kernel:

$${K}_{i,T}\left( x-y\right)=\frac{1}{L\cdot {\text{exp}}\left(\frac{{\left[T-{T}_{i}\right]}^{2}}{2{\sigma }_{i}^{2}}\right)\cdot \sqrt{2\pi }}\cdot {\text{exp}}\left(\frac{{\left[x-y\right]}^{2}}{2\cdot {L}^{2}\cdot {\text{exp}}\left(\frac{{\left[T-{T}_{i}\right]}^{2}}{{\sigma }_{i}^{2}}\right)}\right)$$
(5)

Persistency in shallow waters

Since the O. maya population currently persists in shallow waters of the Yucatan shelf, a traveling wave solution \({N}_{t}\left(s\right)={\text{exp}}\left(c\left(s\right)\cdot s\right)\cdot \phi \left(s\right)\) to the silvered IDE (Eq. 4) is admissible, which resulted in an eigenvalue equation (Lutscher 2019), whose dominant eigenvalue \({\lambda }_{D}\left(s\right)={\text{exp}}\left(c\left(s\right)\cdot s\right)\) and dominant eigenvector \(\phi_D\left(s\right)=\left[\phi_j\left(s\right),\phi_a\left(s\right)\right]^\dagger\)(the \(\dagger\) means transpose operation) allowed calculating the population spread velocity and adult density in shallow waters, respectively. The spread velocity \(c({s}^{*})\) is obtained by clearing \(c(s)\) from \({\lambda }_{D}(s)\) of the traveling wave solution and its subsequent graphical solution, finding the value \({s}^{*}\) of the parameter \(s\) that corresponds to the infimum of \(c\left(s\right)\) (Lutscher 2019):

$$c\left({s}^{*}\right)=\underset{s>0}{{\text{inf}}} \frac{1}{s}\cdot {\text{ln}}{\lambda }_{D}(s)$$
(7)

And the dominant eigenvector defines adult density \({A}_{t}\left(T\right)\) given by the expression:

$${A}_{t}\left(T\right)=\frac{{\phi }_{a}\left({s}^{*}\right)}{{\phi }_{j}\left({s}^{*}\right)+{\phi }_{a}\left({s}^{*}\right)}\times 100\%$$
(8)

Knowing the propagation velocity for each temperature and the distance that the shallow-water O. maya population moves away from the coast each year, its cumulative distance \(D\left(T\right)\) is calculated over time and the spread velocity profile in function of temperature for each scenario proposed by the SSPs. The persistence of the population in shallow water is determined by \({\lambda }_{D}\): the population persists if \({\lambda }_{D}\left(s\right)>1\) and declines when \({\lambda }_{D}\left(s\right)<1\).

Food security of the Yucatan coastal artisanal fishermen

To describe the relationship of O. maya thermal preferences on Yucatan coastal artisanal fishermen’s food security according to the projected temperature increase for SSP1 (+ 3 °C), SSP2 (+ 4 °C), SSP3 (+ 4 °C), SSP4 (+ 4 °C), and SSP5 (+ 5 °C) for 2100, the following indices were constructed: availability \({i}_{\text{av}}\), access \({i}_{\text{ac}}\), utilization \({i}_{\text{ut}}\) complemented with \({\text{AMPI}}\) (average monthly protein indicator) and stability through \({\text{IPI}}\) (income poverty indicator). Although conceptually different, the SSP2, SSP3, and SSP4 scenarios projected the same temperature increase by 2100 therefore compacted as SSP2-4 and analyzed as one for simplicity.

Availability index

Using the growth model developed by Escamilla-Aké et al. (2023), \({i}_{\text{ac}}\) was proposed to quantify O. maya reduction in biomass in shallow waters as temperature increases with respect to the reference temperature of \({T}_{r}=26 ^\circ {\text{C}}\).

According to this model, the individual biomass accumulation as a function of time \(t\) in days and temperature \(T\) in °C is given by:

$$W(t,T) ={W}_{0}\cdot \frac{\left[1+{\text{exp}}\left({\beta }_{T}\cdot {\kappa }_{T}\right)\right]}{\left[1+{\text{exp}}\left({\beta }_{T}\cdot \left[ {\kappa }_{T}-t\right]\right)\right]}\cdot {\text{exp}}\left(\left[{\alpha }_{T}-{\beta }_{T}\right]\cdot t\right)$$
(9)

where \({W}_{0}\) is the initial weight and \({\alpha }_{T}\), \({\beta }_{T},\) and \({\kappa }_{T}\) are empirical temperature functions, which for O. maya are given by:

$${\alpha }_{T}=\alpha \left(T\right)=\frac{0.057}{1+10.5\cdot {\left[\frac{T-26.4}{T+26.4}\right]}^{2}}$$
(10.a)
$${\beta }_{T}=\beta \left(T\right)= \frac{0.051}{1+6.25\cdot {\left[\frac{T-26.4}{T+26.4}\right]}^{2}}$$
(10.b)
$${\kappa }_{T}=\kappa \left(T\right)=\frac{4348}{T}$$
(10.c)

Using this model, the annual relative biomass accumulation rate per organism for any temperature \(T\) projected in SSP with respect to the average reference temperature \({T}_{r}\), during the closed season of duration \(\tau =228 {\text{days}}\) (from December 16 to July 31), allows defining \({i}_{\text{av}}=W\left(\tau ,T\right)/W\left(\tau ,{T}_{r}\right)\):

$${i}_{\text{av}}\left(T\right)=\frac{\left[1+{\text{exp}}\left({\beta }_{T}\cdot {\kappa }_{T}\right)\right]\cdot \left[1+{\text{exp}}\left(\beta {T}_{r}\cdot \left[ {\kappa }_{{T}_{r}}-\tau \right]\right)\right]}{\left[1+{\text{exp}}\left({\beta }_{T}\cdot \left[ {\kappa }_{T}-\tau \right]\right)\right]\cdot \left[1+{\text{exp}}\left({\beta }_{{T}_{r}}\cdot {\kappa }_{{T}_{r}}\right)\right]}\cdot \frac{{\text{exp}}\left(\left[{\alpha }_{T}-{\beta }_{T}\right]\cdot \tau \right)}{{\text{exp}}\left(\left[{\alpha }_{{T}_{r}}-{\beta }_{{T}_{r}}\right]\cdot \tau \right)}$$
(11)

Access index

The \({i}_{\text{ac}}\) proposed depends on the cumulative distances \(D\left(T\right)=\sum_{{T}_{r}}^{T}\left[\left[{c}_{{T}_{k}}\left({s}^{*}\right)\right]\times \left[\text{1 year}\right]\right]\) to which the O. maya stock moves offshore with respect to the typical distances \({L}_{\text{typ}}\approx 20\) km perpendicular to the coast (equivalent to 10 m in depth) that artisanal fishermen travel to catch O. maya and the quasi-income \({f}_{q}\approx 40\%\) of the catch that artisanal fishermen obtain from this fishery:

$${i}_{\text{ac}}\left(T\right)=1-\left[\frac{1-{f}_{q}}{{f}_{q}}\right]\times \left[\frac{D\left(T\right)}{{L}_{\text{typ}}}\right]$$
(12)

where \({s}^{*}\) denotes the particular value of parameter \(s\) that satisfies Eq. (7) for each \({T}_{k}\) from \({T}_{r}\) to \(T\) in SSPs.

Utilization index and average monthly protein indicator

The \({i}_{\text{ut}}\) is affected by O. maya availability and the distance offshore to catch it, so \({i}_{\text{ut}}\) depends directly of \({i}_{\text{av}}\) and inversely of \(D(T)\):

$${i}_{\text{ut}}\left(T\right)={i}_{\text{av}}\left(T\right)\times \frac{{L}_{\text{typ}}}{{L}_{\text{typ}}+D\left(T\right)}$$
(13)

Based on the projection from the year 2020 to the year 2100, of the annual average official catch in \({C}^{\text{kg}}\) kilograms of O. maya from the 2016 to 2020 seasons (https://datos.gob.mx/busca/dataset/produccion-pesquera), \({\text{AMPI}}\) was proposed as a complement of \({i}_{\text{ut}}\). The \({\text{AMPI}}\) describes the average amount of monthly protein that the shallow-water O. maya fishery has the potential to supply during each catch season (from August 1 to December 15).

The \({\text{AMPI}}\) is directly proportional to \({i}_{\text{ut}}\), to the fraction of O. maya utilization for self-consumption \({f}_{u}\approx 10\%\) of the catch, of \({f}_{q}\) and to the catch per community in kilograms \({C}_{T}^{\text{kg}}\), inversely proportional to a fixed number of fishermen in the 2019 official list (https://www.yucatan.gob.mx/padronpesca/) for each community analyzed (number of fishermen): Celestún (1461), Sisal (878), Telchac (310), Dzilam Bravo (1190), San Felipe (522), Rio Lagartos (511), and El Cuyo (669):

$${\text{AMPI}}=\frac{30}{137}\times \frac{{i}_{\text{ut}}\left(T\right)\times {f}_{u}\times {f}_{q}\times {C}_{T}^{\text{kg}}}{{N}_{c}}$$
(14)

The \({\text{AMPI}}\) of the western region is the average \({\text{AMPI}}\) of the communities of Celestún and Sisal; the \({\text{AMPI}}\) of the central region is the average \({\text{AMPI}}\) of the communities of Telchac and Dzilam Bravo; while the \({\text{AMPI}}\) of the eastern region is the average \({\text{AMPI}}\) of the communities of San Felipe, Rio Lagartos, and El Cuyo.

Stability: income poverty indicator

The adjustment line of CONEVAL (Council for the Evaluation of Social Development Policy of Mexico) monthly official rural income poverty data for the period 2000 to 2021 (https://datos.gob.mx/busca/dataset/lineas-de-pobreza-por-ingresos-urbano-rural) allowed the rural income poverty line from 2020 to 2100 for the rural communities analyzed in this research. With these projected values and the O. maya catches per community in Mexican pesos \({C}_{T}^{\mathrm{\$ MXN}}\), the \({\text{IPI}}\) was taken as a stability indicator. The \({\text{IPI}}\) is directly proportional to \({i}_{\text{ut}}\), to the fraction of the catch that is marketed by the fisherman (\(1-{f}_{u}\)), of \({f}_{q}\) and to the catch per community in Mexican pesos \({C}_{T}^{\mathrm{\$ MXN}}\), and inversely proportional to the income poverty line and to the number of fishermen per community:

$${\mathrm{IPI}}=\frac{30}{137}\times \frac{{i}_{\mathrm{ut}}\left(T\right)\times \left[{1-f}_{u}\right]\times {f}_{q}\times {C}_{T}^{\mathrm{\$ MXN}}}{{{\mathrm{VIPL}}\times N}_{c}}$$
(15)

The \({\text{IPI}}\) of the western region is the average \({\text{IPI}}\) of the Celestún and Sisal communities; the \({\text{IPI}}\) of the central region is the average \({\text{IPI}}\) of the Telchac and Dzilam Bravo communities; while the \({\text{IPI}}\) of the eastern region is the average \({\text{IPI}}\) of the San Felipe, Rio Lagartos, and El Cuyo communities Fig. 1.

Fig. 1
figure 1

Study area. Western region: the representative communities of Celestún and Sisal. Central region: the representative communities of Telchac and Dzilam Bravo. Eastern region: the representative communities of San Felipe, Rio Lagartos, and El Cuyo

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Results

The accumulated biomass during the closed season (228 days) at the reference temperature of 26 °C is 2092 g according to Eq. (9) with an initial weight of 0.1 g (Fig. 2). For temperature increases of 1 °C, 2 °C, 3 °C, 4 °C, and 5 °C, the cumulative biomass per animal was 1533 g, 1061 g, 703 g, 452 g, and 286 g (Fig. 2), respectively.

Fig. 2
figure 2

Biomass increase of Octopus maya during the closed season calculated for different temperature increments with respect to the reference temperature (26 °C)

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For the reference temperature, a spread velocity of 0.135 km/year (Fig. 3A) was obtained, while for 1 °C, 2 °C and 3 °C increments with respect to the reference temperature, spread velocities of \(\text{0.181 km/year}\) (Fig. 3B), \(\text{0.299 km/year}\) (Fig. 3C), and \(\text{0.531 km/year}\) (Fig. 3D), respectively. The maximum spread velocity of \(\text{0.586 km/year}\) was obtained for a temperature of 29.4 °C (Fig. 3E). The proportion of adults increases from 4.3% at the reference temperature to a maximum of 5.86% at 29.5 °C; then, it decreases to 4.89% at the temperature of 29.9 °C (Fig. 3F), which marks the temperature limit for population persistence with \({\lambda }_{D}\left({s}^{*}\right)=1\) and \(c\left({s}^{*}\right)=0\) with \({s}^{*}>0\) according to Eq. (7).

Fig. 3
figure 3

Graphical determination of the spread velocity from Eq. (13) for different temperatures: \({\text{A}}\) Reference temperature (26 °C); \({\text{B}}\) +1 °C (27 °C); \({\text{C}}\) +2 °C (28 °C); \({\text{D}}\) +3 °C (29 °C); and \({\text{E}}\) temperature at maximum spread (29.4 °C). \({\text{F}}\) Evolution of the relative proportion of Octopus maya adults in shallow waters

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The spread velocity increases with temperature until it reaches its maximum at 29.4 °C then gradually decreases until it becomes zero at around 29.9 °C (Fig. 4A). For temperatures higher than 29.9 °C, the population stops spreading and declines (\({\lambda }_{D}<1\)) with increasing temperature over time. As the population persists (\({\lambda }_{D}>1\)), the cumulative distance increases with time in the different SSPs. The cumulative distances \(D(T)\) were found to be 21.5 km for 2100 (Fig. 4B), 24.9 km (Fig. 4C), and 20.0 km (Fig. 4D) for SSP1, SSP2-4, and SSP5, respectively. For SSP1, the O. maya population in shallow waters is persistent for 2100 (Fig. 4B). In contrast, the population declines in the years 2098 (Fig. 4C) and 2083 (Fig. 4D) in SSP2-4 and SSP5, respectively.

Fig. 4
figure 4

A Spread velocity profile for different temperatures; \({\text{B}}\) cumulative distance for SSP1 (projected temperature increases of 3 °C); \({\text{C}}\) cumulative distance for SSP2, SSP3, and SSP4 (projected temperature increases of 4 °C) and \({\text{D}}\) cumulative distance for SSP5 (projected temperature increases of 5 °C)

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Food security indices decrease over time. In 2100, the availability index \({i}_{\text{av}}\) (Eq. (11)) decreases to 0.34, 0.22, and 0.14 for SSP1, SSP2-4, and SSP5, respectively (Fig. 5A–C). In the SSP5 for 2085, the shallow water O. maya organisms do not reach the legal catch size (450 g), so from this year onwards, the catches become sublegal (Fig. 5C). The access index \({i}_{\text{ac}}\) (Eq. (12)) decreases to zero in 2081, 2074, and 2068 for SSP1, SSP2-4, and SSP5, respectively (Fig. 5), and consequently, the O. maya fishery becomes unprofitable after these years. In fact, the \({i}_{\text{ac}}\) values begin to be negative, which would imply economic losses for artisanal fishermen on the Yucatan coast by 2100 (Fig. 5).

Fig. 5
figure 5

Projected food security indexes for different SSPs: availability (blue line), access (red line) and utilization (yellow line). \({\text{A}}\) Food security indexes for SSP1 (projected temperature increases of 3 °C); \({\text{B}}\) food security indexes for SSP2, SSP3, and SSP4 (projected temperature increases of 4 °C); and \({\text{C}}\) food security indexes for SSP5 (projected temperature increases of 5 °C)

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The utilization index \({i}_{\text{ut}}\) (Eq. (13)) also decreases along with the availability index \({i}_{\text{av}}\) and as the cumulative distance \(D(T)\) increases for all SSPs. By the year 2100, the utilization rate is reduced to 0.16, 0.10, and 0.07 in SSP1, SSP2-4, and SSP5 respectively (Fig. 5). In addition, the \({\text{AMPI}}\) (Eq. (14)) is reduced in all the region of study for all SSPs; some regions cross the critical line of 2.32 kg of protein per month of the catch season (MCS), indicating that the recommended annual intake (RAI) of 10.6 kg per person (Quaas et al. 2016) may not be reached during the catch season.

For SSP1, the central and eastern regions cross this critical line of 2.32 kg/MCS for 2097 and 2081, respectively (Fig. 6A). Additionally, for SSP2-4, the central and eastern regions cross the critical line for 2092 and 2079, respectively (Fig. 6C). Finally, for the SSP5, all the regions cross the critical line: the western region for 2092, central region for 2080, and eastern region for 2070 (Fig. 6E).

Fig. 6
figure 6

Projected average monthly protein indicator (Eq. (14 )\(\text{)}\) to year 2100, from the Octopus maya average catch in kilograms during the 2016 to 2020 catch seasons for each region and according to \({\text{A}}\) SSP1, \({\text{C}}\) SSP2-4, and \({\text{E}}\) SSP5. Projected income poverty indicator (Eq. (15)) to year 2100, from the Octopus maya average catch in Mexican pesos from 2016 to 2020 catch seasons for each region and the projected income poverty line from the adjusted line to incomer poverty data from year 2000 to year 2021, and according to \({\text{B}}\) SSP1, \({\text{D}}\) SSP2-4, and \({\text{F}}\) SSP5

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The stability index that corresponds with \({\text{IPI}}\) (Eq. (15)) decreases in all the regions studied and falls below the income poverty line defined by the CONEVAL. According to each of the SSPs, the years in which the regions cross the income poverty line are as follows: western region with SSP1, SSP2-4, and SSP5, respectively, in 2058, 2053, and 2050; central region with SSP1, SSP2-4, and SSP5, respectively, in 2052, 2048, and 2045; and finally, the eastern region with SSP1, SSP2-4, and SSP5, respectively, in 2042, 2040, and 2038 (Fig. 6B, D, F).

Discussion

Increased catches may mask the food security impact of rising temperatures in shallow waters

Considering the biomass reduction predicted by Eqs. (9) and (10a, b, c) for temperatures above the reference temperature of 26 °C, negative impacts can be predicted on the artisanal fishery of O. maya in shallow waters of the Yucatan shelf (Fig. 2). On the other hand, temperatures above 26 °C tend to favor the proliferation of adults in shallow waters, increasing their proportion compared to that of juveniles, which decrease as temperature increases (Fig. 3F). Given that O. maya adults tend to be the most caught by the authorized fishing gear (“gareteo”), it would appear that temperatures above 26 °C favor their capture in shallow waters. However, this apparent benefit in catches (Arreguín-Sánchez 2019) hides the negative impact that the increase in temperature has and will have on the O. maya population in shallow waters due to climate change and food security of artisanal fishermen from the Yucatan coast. This negative impact can be better understood by analyzing its repercussion on each of the dimensions of food security: availability, access, utilization, and stability (FAO 2008).

Availability

The evolution of the \({i}_{\text{av}}\) in the different SSPs suggests a reduction in O. maya availability as food for artisanal fisheries despite the fact that high temperatures also favor its legally catchable stock. This result is similar to what was mentioned in Arreguín-Sánchez (2019) but with two fundamental differences.

The first one is that favoring the legally catchable O. maya population in shallow waters is due to the increase in the adult proportion as temperature increases due to climate change. This situation contrasts with what Arreguín-Sánchez (2019) mentioned about a supposed benefit of high temperatures for O. maya growth. In fact, as mentioned here and suggested by different studies (Juárez et al. 2015; López-Galindo et al. 2019; Meza-Buendía et al. 2021), temperatures above 26 °C negatively affect O. maya fitness and in particular its biomass (Escamilla-Aké et al. 2023). The second different aspect is that this increase will peak around 29.5 °C and then decrease. In fact, from 29.9 °C onward, the entire shallow water population will decline (Fig. 4A).

Finally, although the O. maya population in shallow waters is maintained and legally catchable for the SSP1 scenario until 2100, it is alarming that higher and sustained temperature increases, such as those described in SSP2-5, comprise not only the O. maya population persistence and its availability in shallow waters, but also its legality in terms of catches possibly due to natural temperature fluctuation before 2100 (Liu et al. 2015). In this sense, the present research shows that for increases of 4 °C with respect to the reference temperature, O. maya populations barely may reach the minimum of 450 g for legal catch (SSP2-4) in 2100, while increases of 5 °C leave O. maya catches in shallow waters in the realm of sub-legality (< 450 g) from the year 2085 onwards.

Access

As temperature increases with respect to the reference temperature, the results of the present research study suggest that the persistent shallow-water O. maya population (\({\lambda }_{D}>1\)) spreads with increasing velocity until it reaches a maximum of 0.596 km/year at 29.4 °C. Then, it decreases to become zero around 29.9 °C; the temperature at which the shallow-water O. maya population begins to decrease (\({\lambda }_{D}<1\)) as the temperature continues to increase with time.

Temperatures above 26 °C but below 29.5 °C favor the proliferation of adults in shallow waters but reduce the individual biomass and increase the distance to shore that artisanal fishermen have to travel to catch O. maya. In addition, for temperatures above 29.5 °C, the adult population gradually declines, and at 29.9 °C, the entire population begins to decline. Consequently, the access of O. maya as food in shallow waters decreases as temperature increases over time, as indicated by the \({i}_{\text{ac}}\), since greater distances imply a greater physical and economic effort for its capture. This result is reflected in the gradual reduction of the quasi-rent obtained by artisanal fishermen of the Yucatan coast for the capture of this species in shallow waters. The distances accumulated by the O. maya population in all the SSPs analyzed was around 20 km by 2100, which is compatible with the general trend of various marine organisms in the world to spread toward colder areas due to the increase in temperature (Poloczanska et al. 2013).

In this context, the shallow-water O. maya fishery will cease to be profitable for artisanal fishermen in 2081 for SSP1, in 2074 for SSP2-4, and in 2068 for SSP5 and will generate economic losses for these fishermen after these years.

Utilization

The index \({i}_{\text{ut}}\) for O. maya as food shows a decreasing trend as temperature increases with time in the different SSPs. A more pronounced reduction is observed with respect to the availability index \({i}_{\text{av}}\), as \({i}_{\text{ut}}\) varies inversely with the distance accumulated by the persistent shallow-water O. maya population moving away from the coast with increasing temperature over time.

Note that the greater the availability of the resource is, the easier the utilization of O. maya for consumption becomes as long as the economic or physical cost, associated with the distance of capture, is adequate; as the cost increases, utilization decreases even if the resource is available. To complement this information, \({\text{AMPI}}\) describes the average monthly amount of protein that the shallow-water O. maya fishery would be able to provide in the catch season to artisanal fishermen in the western, central, and eastern Yucatan shelf regions in each of the hypothetical SSPs.

Assuming the minimum temperature increase of SSP1, O. maya shallow water catches will not be able to supply the RAI of 10.6 kg of protein per person from 2081 for the eastern region and from year 2097 for the central region. Similarly, for SSP2-4, the eastern and central regions will not be able to supply the RAI of 10.6 kg of protein per person from 2079 and 2092, respectively. In fact, for SSP5, none of the analyzed regions will have this capacity starting some year before the year 2100.

Stability

In Mexico, the basic food basket added to the non-food basket constitutes the income poverty line, whose trend continues to rise. In this sense, if O. maya catches in shallow waters are reduced, both in terms of availability and physical or economic access to this resource, artisanal fishermen who depend mainly on this fishery could find themselves in an increasing state of food vulnerability.

The proposed \({\text{IPI}}\) indicator reveals this trend for the analyzed regions regardless of the SSP chosen. All the analyzed regions will reach \({\text{IPI}}\) values below the food poverty line as temperature increases over time. Moreover, if the catch is restricted exclusively to this species, regardless of the SSP, in none of the regions analyzed will the shallow water O. maya fishery be able to provide sufficient resources to support the families of artisanal fishermen by 2030 and neither fisherman alone by 2060.

Assumptions and work limitations

Although the methodology developed in this research is general, it depends on the quality of the data available for calculating the magnitudes of interest, such as vital rates, spread speed, cumulative distance, indices, and food security indicators. Moreover, in the case of vital rates assumed to be proportional to the probability density of each substage presence, empirical estimates were made thanks to the existence of the experimental data on O. maya preferred temperatures in each stage and the ranges that have shown to be favorable for their development.

With respect to spread speed and cumulative distance, the scale was based on the density estimation of the organisms per square kilometer through the parameter \(L\). Considering higher O. maya densities per square kilometer, \(L\) increases, which results in an increase in spread speed and cumulative distance. Therefore, given the low density used (Wakida Kusunoki et al. 2004), the present research study could be underestimating these quantities. Consequently, the negative effects of temperature increase on O. maya wild population and food security of the artisanal fishermen who catch this species in shallow waters could be much greater and occur in a shorter time than predicted.

Finally, although \({\text{AMPI}}\) and \({\text{IPI}}\) indicators are based on a projection of official data and such data may raise considerable doubts about their reliability, they are still useful as prescriptive rather than predictive tools, as they provide us with information on the future trends and challenges that artisanal fishers in the coastal regions may face in terms of food security.

Conclusions

This research study made O. maya thermal preferences operational to analyze its shallow water population growth and offshore propagation. The increased temperature impact on the species population was evaluated, including the food security of the artisanal fishermen that catch this species in shallow waters in the western, central, and eastern regions of the Yucatan shelf.

The results obtained suggest that the possible increase in catches in the coming years may be due to an increase in the proportion of adults and not to a supposed benefit of high temperatures for O. maya development. In fact, this possible increase in future catches may mask the negative impact that rising temperatures may have on the wild populations of this species in shallow waters and, consequently, the negative impacts on food security.